diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkjno" "b/data_all_eng_slimpj/shuffled/split2/finalzzkjno" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkjno" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\\noindent The problem of wealth allocation is a key driver for past and present economic studies. Since the seminal contribution of \\cite{markowitz.1952}, both the industry and the researchers, have been highly involved in finding the optimal way to improve the performance of allocation strategies. During last decades, the Markowitz's approach has been widely criticised for the assumptions behind its simple mean--variance framework. In particular, the most critical points regard the homoskedasticity assumption and, in general, the Gaussian assumption for the joint returns distribution.\nRecently, researchers and practitioners found that the departure from the simple Markowitz's mean--variance approach, considering, for example, higher moments, can provide substantial improvements of the asset allocation strategies in terms of cumulative returns, see, e.g., \\cite{jondeau_rockinger.2003}, \\cite{lai_etal.2006}, \\cite{harvey_etal.2010} and \\cite{holly_etal.2011}. Furthermore, as for the conditional second moment, there have also been several attempts to model the conditional skewness and kurtosis of financial time series.\nIndeed, starting from the seminal contribution of \\cite{hansen.1994}, in the last two decades, the financial econometric literature has been focusing on modelling time--varying higher--order moments. One of the main issues is to figure out the correct way to introduce time--variation in skewness and kurtosis. Most important contributions in this sense are those of \\cite{hansen.1994}, \\cite{theodossiou_1998}, \\cite{harvey_siddique.1999}, \\cite{jondeau_rockinger.2003}, \\cite{deluca_etal.2006} and \\cite{jondeau_rockinger.2012}, who extend the model of \\cite{hansen.1994} and the class of ARCH models of \\cite{engle.1982} and \\cite{bollerslev.1986}. However, when dealing with higher--order conditional moments' dynamics in a multivariate context, the complexity increases principally for the high number of parameters needed and the major concern involves the overall tractability of the resulting model. As regards the co--skewness, main results are referred to \\cite{deluca_etal.2006}, \\cite{deluca_loperfido.2012} and \\cite{franceschini_loperfido.2010}, while for the co--kurtosis, up to our knowledge, there are not any contributions that impose a conditional dynamics on the fourth moment.\\newline\n\\indent The problem of dealing with higher order co--moments is strictly related to the general issue of describing the dependence structure affecting the multivariate financial time series. Since the seminal contribution of \\cite{sklar.1959}, copula models have been effectively used as an alternative flexible tool to the classical multivariate Gaussian models. Indeed, in the last years, copulas theory has played a central role for the financial econometric literature, as widely documented by \\cite{mcneil_etal.2005} and \\cite{joe.2014}. A comprehensive treatments of recent developments in this field can be found in the recent book of \\cite{durante_sempi.2015}. Furthermore, starting from the seminal contribution of \\cite{patton.2006a} and \\cite{jondeau_rockinger.2006b}, copula theory has been further extended to cope with dynamic evolving dependence structures. Time--varying copulas have been successfully employed in empirical works such as in \\cite{rodriguez.2007}, \\cite{delirasalvatierra_patton.2015} and \\cite{bernardi_catania.2015a}.\\newline\n\\indent An effective way to improve the models in sample fit as well as their forecast ability, is to include exogenous information. Nevertheless, most econometric models that aim to explain the joint dependence between economic factors, only consider endogenous information. By the way, it is natural to believe that, the dynamics affecting the relations between industries, and more generally between economies, are highly affected by exogenous, as well as, endogenous factors. It is worth noting that, the economic and financial literature focusing on univariate models, makes extensive use of exogenous information, see, e.g., \\cite{pagan.1971} and \\cite{baillie_1980} in the ARMAX context and \\cite{hwang_satchell.2005} for the GARCHX volatility model. Their empirical results confirm the superior ability of the models exploiting exogenous information, as compare with more simple alternatives. However, despite the documented usefulness of accounting for exogenous information in univariate time series models, in the financial econometric literature that focuses on the dynamic dependence modelling, there are only few examples of multivariate models dealing with exogenous information, see, e.g., \\cite{delirasalvatierra_patton.2015}. This lack is principally related to the difficulty of effectively including such information in such a way to retain model tractability, especially when the dimension of the observed variables is greater than 2. However, the contribution of including exogenous information in order to describe the dependence structure of economies or assets cannot be neglected. Let think about how the recent Global Financial Crises (GFC) of 2007--2008 influenced the dependence between economies. It is undoubtedly true that such a dramatic event drastically changed the choices of wealth allocation of economic agents. Indeed, capitals have moved away from the financial sector, and more generally from speculative instruments, such as options and futures, opening positions on less riskier assets such as short term government bonds or liquidity funds. In this reallocation of resources, it is obvious that, the behaviour of the sovereign interest rates dynamics, for examples, plays a central role in determining how much of the total wealth is reallocated, and hence how the dependence structure of the remaining assets is affected.\\newline\n\\indent In this paper, we propose a new Markov switching dynamic copula model being also able to deal with exogenous information in a fully multivariate setting using elliptical copulas. More precisely, the tractability induced by the copula methodology in dealing with marginal and joint separability is gathered with a new dynamic conditional dependence structure that relies on the presence of exogenous covariates and the inclusion of a latent Markovian structure, in order to enhance the model ability to account for time--varying higher moments. In this way, we are able to incorporate relevant information such as third and fourth conditional moments, as well as economic factors, in a portfolio optimisation framework. Throughout the paper, we will distinguish the marginal modelisation from the dependence structure linking individual asset into a multivariate framework. This distinction is available thanks to the use of copulas, and more precisely, to the \\cite{sklar.1959} Theorem and to the Inference Function for Margins (IFM) technique of \\cite{godambe_1960} and \\cite{mcleish_small.1988}.\nAs regards the univariate specification, we develop a new flexible parametric model accounting for time--varying conditional moments up to the fourth of the univariate distribution. In particular, we rely on the Generalised Autoregressive Score (GAS), also known as Dynamic Conditional Score (DCS), framework recently proposed by \\cite{creal_etal.2013} and \\cite{harvey.2013} in order to update the dynamics of time--varying parameters using the scaled score of the conditional distribution. This specification allows us to take account of a variety of well know stylised facts affecting the financial time series, like time varying moments, skewness and excess of kurtosis, as widely discussed by \\cite{mcneil_etal.2005}. Particularly interesting is the ability of the employed univariate model to approximate the unknown dynamics of the conditional density of returns by means of a filter based on the scaled score of the conditional distribution. This approach has been recently proved to be consistent with the goal of describing time variation for densities' parameters, see \\citep{koopman_etal.2015}.\n\\indent As stated before, the multivariate modelisation is carried on by using a copula function that links the dependence structure of each univariate series. This solution has been proved to provide highly flexible and reliable models for multivariate financial time series analysis, see e.g. \\cite{patton.2006b}, \\cite{joe.2014}, and the references quoted therein. Moreover, the seminal work of \\cite{patton.2006a}, also extend the original IFM two step estimator to the time--varying copula parameters framework. This fundamental result allows to effectively model the time--varying behaviour of linear as well as non--linear dependence patterns of financial returns. As discussed by \\cite{mcneil_etal.2005}, the flexibility in modelling linear and non--linear dependence patterns is one of the main stylised fact affecting multivariate financial time series. Indeed, as a direct consequence, econometric models dealing with multivariate financial time series, are required to account for this aspect, see, e.g., \\cite{delirasalvatierra_patton.2015}. For this reason, we propose a new specification for the evolution of the conditional correlation matrix of elliptical copulas relying on the famous Dynamic Conditional Correlation (DCC) specification of \\cite{engle.2002} and \\cite{tse_tsui.2002}, and the extension of \\cite{cappiello_etal.2006}. Moreover, as in \\cite{billio_caporin.2005} and, more recently, \\cite{bernardi_catania.2015a}, we also allow for the dynamic evolution of the conditional correlation matrix to depend upon the realisation of a first order Markovian process. More precisely, our model allows for different dynamic behaviours of the underlying dependence process in each specific state of the nature, as in \\cite{billio_caporin.2005} and \\cite{bernardi_catania.2015a}, but it additionally includes exogenous regressors in the dependence structure conditional to the latent state. The proposed model can effectively describe tranquil as well as turbulent periods affecting the financial markets, as for example, different phases of the economic or financial cycles. The opportunity to include exogenous covariates into the correlation dynamics is motivated by the observation that, in nowadays financial markets, economic agents usually base their decisions about wealth allocation conditionally on a huge amount of available information coming from multiple sources, which cannot be neglected. Summarising, our new specification is specifically tailored to model time variation in higher order conditional moments and possibly nonlinear time variation in the dependence structure of financial time series including the ability to account for exogenous information.\\newline\n\\indent The empirical part of the paper considers a panel of financial returns in a portfolio optimisation framework using higher order moments of the predictive conditional joint distribution. More precisely, in a similar way as \\cite{jondeau_rockinger.2006a}, we consider conditional portfolio moments into the expected utility of a rational investor by means of a Taylor expansion around the future wealth. Our empirical findings support our model since it outperform famous alternatives, like the DCC model. Specifically, the new specification improves the performance of allocation strategies based on the maximisation of the expected utility.\\newline\n\\indent The remaining of the paper is organised as follows. Section \\ref{sec:model} introduces the marginal and the joint dynamic model specifications and discusses their main features. Section \\ref{sec:asset_allocation} describes the asset allocation problem. Section \\ref{sec:empirical_study} deals with the empirical study and Section \\ref{sec:conclusion} concludes.\n\\section{The Model}\n\\label{sec:model}\n\\noindent In this paper, we propose to estimate the parameters governing the copula function and the associated dependence structure separately from those controlling for the dynamic evolution of the marginal distributions. This estimation technique, known as Inference Function for Marginals (IFM), has been proved to be feasible in large parametric spaces and asymptotically consistent for time varying copulas, see, e.g., \\cite{patton.2006a}. One of the main advantages of using the IFM estimating procedure concerns its ability to separate the univariate financial returns stylised facts from those regarding the multivariate distribution. This latter aspect, allows the researcher to focus on multivariate dependence structure only in a second moment, when the most appropriate model to describe all the empirical regularities of the observed univariate series has been selected and estimated. Whenever the marginal models are able to capture all the univariate empirical regularities of the marginal series, the only interestingly phenomenon to study is the remaining dependence structure. It follows that, the choice of an appropriate model that acts as a filter for the univariate series plays an important role for the subsequent multivariate analysis, see, e.g., \\cite{mcneil_etal.2005}.\\newline\n\\subsection{Marignal models}\n\\label{sec:marginal_model}\n\\noindent The financial econometrics literature offers several valid alternatives to model the conditional distribution of financial returns. In a fully parametric setting, following the same arguments presented by \\cite{cox.1981}, the first issue is to choose between the classes of observation driven and parameter driven models. The former can be well represented by the GARCH family of \\cite{engle.1982} and \\cite{bollerslev.1986}, while the latter is typically associated with the class of state space models; see for example \\cite{harvey_proietti.2005} and \\cite{durbin_koopman.2012}. However, the Generalised Autoregressive Score (GAS) framework, recently introduced by \\cite{creal_etal.2013} and \\cite{harvey.2013} is gaining lots of consideration by econometricians in many fields of time series analysis. Under the \\cite{cox.1981} classification, the GAS models can be considered as a class of observation driven models, with the usual consequences of having a closed form expression for the likelihood and ease of evaluation. However, as noted by \\cite{koopman_etal.2015}, the class of GAS models is similar to the class of state space models in the sense that both approaches make use of the information coming from the entire conditional distribution instead of using only that coming from the conditional expected value, as for example in the GARCH framework.\nIndeed, the key feature of GAS models is that the score of the conditional density is used as the forcing variable into the updating equation of a time--varying parameter. In the recent financial and econometrics literature, many reasons have been argued to support the adoption of score--based rules as a general updating mechanism of time--varying parameters, see, e.g., \\cite{harvey.2013} and \\cite{creal_etal.2013}, just to quote a few of them. Moreover, the flexibility of the GAS framework makes this class of models nested with a huge amount of famous econometrics models such as, for example, some of the ARCH--type models of \\cite{engle.1982} and \\cite{bollerslev.1986} for volatility modelling, and also the MEM, ACD and ACI models of \\cite{engle.2002b}, \\cite{engle_gallo.2006}, \\cite{engle_russell.1998} and \\cite{russell.1999}, respectively. Finally, one of the practical implications of using this framework in order to update the time--varying parameters, is that it avoids the problem of using a non--adequate forcing variable when the choice of it is not so obvious as for dynamic copula models; see, e.g., \\cite{bernardi_catania.2015a}.\\newline\n\\indent Formally, we assume that the $i$--th return at time $t$, for $t=1,2,\\dots,T$, is conditionally distributed according to the filtration $\\mathcal{F}_{i,t-1}=\\sigma\\left(y_{i,1},y_{i,2},\\dots,y_{i,t-1}\\right)$ as\n\\begin{equation}\ny_{i,t}\\sim\\mathcal{AST}\\left(y_{i,t};\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\mid\\mathcal{F}_{t-1}\\right),\n\\end{equation}\nwhere $\\mathcal{AST}\\left(y_{i,t};\\cdot\\right)$ denotes the Asymmetric Student--t distribution with equal left and right tail decay of \\cite{zhu_galbraith.2010} with time varying location $\\mu_{i,t}\\in\\mathbb{R}$, scale $\\sigma_{i,t}\\in\\mathbb{R}^+$, shape $\\nu_{i,t}\\in\\left(4,\\infty\\right)$ and skewness $\\gamma_{i,t}\\in\\left(0,1\\right)$ parameters and density given by\n\\begin{equation}\nf_\\mathcal{AST}\\left(y_{i,t};\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)=\\begin{cases}\n\\frac{1}{\\sigma_{i,t}}\\left[1+\\frac{1}{\\nu_{i,t}}\\left(\\frac{y_{i,t}-\\mu_{i,t}}{2\\gamma_{i,t}\\sigma_{i,t} K\\left(\\nu_{i,t}\\right)}\\right)\\right]^{-\\frac{\\left(\\nu_{i,t}+1\\right)}{2}},\\quad& y_{i,t}\\le\\mu_{i,t}\\nonumber\\\\\n\\frac{1}{\\sigma_{i,t}}\\left[1+\\frac{1}{\\nu_{i,t}}\\left(\\frac{y_{i,t}-\\mu_{i,t}}{2(1-\\gamma_{i,t})\\sigma_{i,t} K\\left(\\nu_{i,t}\\right)}\\right)\\right]^{-\\frac{\\left(\\nu_{i,t}+1\\right)}{2}},\\quad& y_{i,t}>\\mu_{i,t},\\nonumber\\\\\n\\end{cases}\n\\end{equation}\nwhere $K\\left(x\\right) = \\Gamma\\left(\\left(x+1\\right)\/2\\right)\/\\left[\\sqrt{\\pi x}\\Gamma\\left(x\/2\\right)\\right]$ and $\\Gamma\\left(\\cdot\\right)$ is the gamma function. We also define $\\mathbf{\\theta}_{i,t} = \\left(\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)^\\prime\\in\\mathbb{R}\\times\\mathbb{R}^+\\times\\left(0,1\\right)\\times\\left(4,\\infty\\right)$ to be a vector containing all the time--varying parameters. Furthermore, let $h:\\mathbb{R}^4\\to\\mathbb{R}\\times\\mathbb{R}^+\\times\\left(0,1\\right)\\times\\left(4,\\infty\\right)$ be a continuous, $\\mathcal{F}_{t-1}$ measurable, twice differentiable, vector--valued mapping function, such that $\\mathbf{\\theta}_t = h\\left(\\tilde{\\mathbf{\\theta}}_{i,t}\\right)$, where $\\tilde\\mathbf{\\theta}_{i,t}\\in\\mathbb{R}^4$ is a proper reparametrisation of $\\mathbf{\\theta}_{i,t}$, for all $t=1,2,\\dots,T$. In our context, a convenient choice for $h\\left(\\cdot\\right)$ is\n\\begin{equation}\nh\\left(\\tilde{\\mathbf{\\theta}}_{i,t}\\right) = \\begin{cases}\n\\mu_{i,t}=\\mu_{i,t}\\\\\n\\sigma_{i,t}=\\exp\\left(\\tilde\\sigma_{i,t}\\right)\\\\\n\\gamma_{i,t}=\\left[1 + \\exp\\left(-\\tilde\\gamma_{i,t}\\right)\\right]^{-1}\\\\\n\\nu_{i,t}=4+\\exp\\left(\\tilde\\nu_{i,t}\\right).\n\\end{cases}\n\\end{equation}\nWe let the time--varying reparameterised vector $\\tilde\\mathbf{\\theta}_{i,t}=\\left(\\mu_{i,t},\\tilde\\sigma_{i,t},\\tilde\\gamma_{i,t},\\tilde\\nu_{i,t}\\right)^\\prime$ to be updated using the score of the conditional distribution of $y_{i,t}$, exploiting the GAS dynamic where the Fisher information matrix of the conditional distribution is used as scaling matrix, as suggested by \\cite{creal_etal.2013} and \\cite{harvey.2013}. Specifically, we consider the following updating mechanism for the parameter dynnamics\n\\begin{align}\n \\begin{pmatrix}\n \\mu_{i,t+1}\\\\\n \\tilde\\sigma_{i,t+1}\\\\\n \\tilde\\gamma_{i,t+1}\\\\\n \\tilde\\nu_{i,t+1}\n \\end{pmatrix} =\n \\begin{pmatrix}\n \\omega_{\\mu_i}\\\\\n \\omega_{\\sigma_i}\\\\\n \\omega_{\\gamma_i}\\\\\n \\omega_{\\nu_i}\n \\end{pmatrix} +\n \\begin{pmatrix}\n \\alpha_{\\mu_i} & 0 & 0 & 0\\\\\n 0 & \\alpha_{\\sigma_i} & 0 & 0\\\\\n 0 & 0 & \\alpha_{\\gamma_i} & 0 \\\\\n 0 & 0 & 0 & \\alpha_{\\nu_i}\n \\end{pmatrix} \\tilde{s}_{i,t} +\n \\begin{pmatrix}\n \\beta_{\\mu_i} & 0 & 0 & 0\\\\\n 0 & \\beta_{\\sigma_i} & 0 & 0\\\\\n 0 & 0 & \\beta_{\\gamma_i} & 0 \\\\\n 0 & 0 & 0 & \\beta_{\\nu_i}\n \\end{pmatrix}\n \\begin{pmatrix}\n \\mu_{i,t}\\\\\n \\tilde\\sigma_{i,t}\\\\\n \\tilde\\gamma_{i,t}\\\\\n \\tilde\\nu_{i,t}\n \\end{pmatrix},\n \\label{eq:gas_empDyn}\n\\end{align}\nwhere $\\tilde s_t = \\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right)^{-1}\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right)^{-1}\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ is the scaled score of the reparameterised conditional distribution of $y_{i,t}$. The quantity $\\tilde s_t$ depends on the product of three matrices defined as\n\\begin{align}\n\\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right) &= \\frac{\\partial\\mathrm{h}\\left(\\tilde{\\boldsymbol{\\theta}}_{i,t}\\right)}{\\partial\\tilde\\mathbf{\\theta}_{i,t}}\\\\\n\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right) & = \\mathbb{E}\\left[\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)\\times\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)^\\prime\\mid\\mathcal{F}_{t-1}\\right]\\\\\n\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right) &= \\frac{\\partial\\ln f_\\mathcal{AST}\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)}{\\partial{\\mathbf{\\theta}_{i,t}}},\n\\end{align}\nwhere $\\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right)$ is the Jacobian of the mapping function $h\\left(\\cdot\\right)$, i.e.\n\\begin{equation}\n \\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right) = \\begin{pmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & \\exp\\{\\tilde\\sigma_{i,t}\\} & 0 & 0 \\\\\n 0 & 0 & -\\exp\\{-\\tilde\\gamma_{i,t}\\}\\left(1+\\exp\\{-\\tilde\\gamma_{i,t}\\}\\right)^{-2} & 0 \\\\\n 0 & 0 & 0 & \\exp\\{\\tilde\\nu_{i,t}\\},\n \\end{pmatrix},\n\\end{equation}\nwhere $\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right)$ and $\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ are the Fisher information matrix and the score of the conditional AST distribution, respectively. For the parameterisation here considered, the Fisher information matrix of the AST distribution is reported in \\cite{zhu_galbraith.2010}, while the score is reported in \\ref{sec:ScoreAndFisher}.\\newline\n\\indent It is worth noting that, in equation \\eqref{eq:gas_empDyn}, a diagonal structure for the matrix of coefficient is considered. Obviously, this is a convenient choice in order to reduce the number of model parameters, but other solutions are possible. In order to guarantee the stationarity of the process defined in equation \\eqref{eq:gas_empDyn} we only need to impose the constraint that all the autoregressive coefficients $\\beta_k$, $k\\in\\left(\\mu,\\sigma,\\gamma,\\nu\\right)$ would be in modulus less then one, since, under a correct model specification, the sequence of scores $\\{\\tilde s_t, t>0\\}$ is a martingale difference.\n\\subsection{Multivariate model}\n\\label{sec:multivariate_model}\n\\noindent Dynamic time--varying dependence structures of financial assets have been deeply investigated in the recent developments of financial econometrics. One of the most successful and powerful model has been the Dynamic Conditional Correlation (DCC) model of \\cite{engle.2002} and \\cite{tse_tsui.2002} which allows for time variation in the correlation of assets returns. In the last few years many models were also been proposed in order to improve the flexibility of the original DCC, see, e.g., \\cite{bauwens_etal.2006} for a complete surveys of these recent developments. The success of the DCC model is due to its two stage estimation procedure which mainly relies on the multivariate Gaussian assumption. However, when the Gaussian assumption is violated, the separability of the log likelihood does not hold anymore and the only viable alternative in order to remain with a feasible model is to adopt a Quasi--MLE estimation approach, see, e.g., \\cite{engle_sheppard.2001}. Despite its enormous popularity, only few works have been focused on the inclusion of exogenous information in a DCC framework in order to let the covariance co--movements to depend on covariates. Among those, \\cite{vargas.2008} firstly introduced exogenous covariates into the Asymmetric Generalised DCC (AGDCC) model of \\cite{cappiello_etal.2006}, while \\cite{chou_liao.2008} follow a similar approach on a simpler DCC model. However, both the empirical applications of the aforementioned papers, only consider a bivariate setting under the Gaussian assumption for the innovation terms. Besides the tractability of the resulting model, one of the main difficulties of including exogenous covariates into a DCC model, is to guarantee the positiveness of the variance--covariance matrix at each point in time. A common solution relies on the Constrained Maximum Likelihood (CML) or Penalised Maximum Likelihood (PML) estimation approaches. In the context of bivariate time--varying copulas, only \\cite{delirasalvatierra_patton.2015} consider the inclusion of exogenous information to model the correlation matrix dynamic. However, \\cite{delirasalvatierra_patton.2015} do not consider a DCC--type dynamic evolution of the copula parameters, but, they instead rely on the GAS framework of \\cite{creal_etal.2013} and \\cite{harvey.2013}. Moreover, their model cannot be used in a fully multivariate setting when considering elliptical copulas such as the Gaussian and the Student--t. In what follows, we introduce our Student--t dynamic Markov--Switching (MS) copula model where the copula dependence matrix evolves according to a MS--DCC model which also depends on exogenous covariates.\\newline\n\\indent Let $\\mathbf{u}_t=\\left(u_{1,t},u_{2,t},\\dots,u_{N,t}\\right)^\\prime$ with $u_{j,t}=F_{\\mathcal{AST}}\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ be the Probability Integral Transformation (PIT) of $y_{i,t}$ according to the conditional AST distribution $F_{\\mathcal{AST}}\\left(\\cdot\\right)$ with parameters $\\mathbf{\\theta}_{i,t}=\\left(\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)^\\prime$, and assume that\n\\begin{equation}\n\\mathbf{u}_t\\mid S_t=s\\sim\\mathcal{C}_T\\left(\\mathbf{u}_t;\\mathbf{R}_t^s,\\nu_\\mathsf{c}^s\\right),\n\\end{equation}\nfor $t=1,2,\\dots,T$, where $\\mathbf{R}_t^s$ is the time--varying, regime dependent correlation matrix at time $t$ of the pseudo--observation $\\mathbf{u}_t$, and $\\nu_\\mathsf{c}^s$ is the degree of freedom parameter that is also subject to the realisation of the first order Markov chain. According to the DCC dynamics the conditional correlation matrix $\\mathbf{R}_t^s$ can be decomposed in the following way\n\\begin{align}\n\\mathbf{R}_t^s={\\mathbf{D}_t^s}^{-1}\\mathbf{C}_t^s{\\mathbf{D}_t^s}^{-1},\n\\end{align}\nwhere $\\mathbf{D}_t^{s}$ is a diagonal matrix containing the square root of the diagonal elements of $\\mathbf{C}_t^s$. We further assume that the latent states $s=1,2,\\dots,L$ are driven by a Markov process $S_t$, for $t=1,2,\\dots,T$ defined on the discrete space $\\Omega=\\left\\{1,2,\\dots,L\\right\\}$ with transition probability matrix $\\mathbf{Q}=\\left\\{q_{l,k}\\right\\}$, where $q_{l,k}=\\mathbb{P}\\left(S_t=k\\mid S_{t-1}=l\\right)$, $\\forall l,k\\in\\Omega$ is the probability that state $k$ is visited at time $t$ given that at time $t-1$ the chain was in state $l$, and initial probabilities vector $\\mbox{\\boldmath $\\delta$}=\\left(\\delta_1,\\delta_2,\\dots,\\delta_L\\right)^\\prime$, $\\delta_l=\\mathbb{P}\\left(S_1=l\\right)$, i.e., the probability of being in state $l=\\left\\{1,2,\\dots,L\\right\\}$ at time 1. For an up to date review of HMMs, see, e.g., \\cite{cappe_etal.2005}, \\cite{zucchini_macdonald.2009} and Dymarski \\citeyearpar{dymarski.2011}. The following dynamics is imposed on the state dependent variance--covariance matrix\n\\begin{align}\n\\mathbf{C}_{t+1}^s&= \\left(\\bar{\\mathbf{C}}-\\mathbf{A}^s\\bar{\\mathbf{C}}{\\mathbf{A}^{s}}^\\prime-\\mathbf{B}^s\\bar{\\mathbf{C}}{\\mathbf{B}^{s}}^\\prime-\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\bar{\\mathbf{X}}\\right)\n+\\mathbf{A}^s\\bXi_t{\\mathbf{A}^{s}}^\\prime+\\mathbf{B}^s\\mathbf{C}_t^s{\\mathbf{B}^{s}}^\\prime+{\\mathbf{K}\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\mathbf{X}_t,\n\\label{eq:corr_dynamic}\n\\end{align}\nfor $s=1,2,\\dots,L$, where $\\mathbf{A}^s$ and $\\mathbf{B}^s$ are diagonal matrices with elements $\\mathbf{A}^s=\\left\\{\\sqrt{\\alpha_i^s}\\right\\}$, $\\mathbf{B}^s=\\left\\{\\sqrt{\\beta_i^s}\\right\\}$, for $i=1,2,\\dots,N$, $s=1,2,\\dots,L$, $\\mathbf{K}$ is a $N\\times N$ matrix of ones, $\\mbox{\\boldmath $\\xi$}^s$ is a $p\\times 1$ vector containing the coefficients associated to the exogenous variables $\\mathbf{X}_t$. Here, $\\bar\\mathbf{C}$ denotes the empirical variance--covariance matrix of the pseudo--observations $\\mathbf{u}_t$ and $\\bar\\mathbf{X}$ is a $p\\times 1$ vector containing the empirical mean of the regressors, which are introduced in equation \\eqref{eq:corr_dynamic} in order to enforce the mean--reversion of the process. Moreover, in order to guarantee the stationarity of the process defined in equation \\eqref{eq:corr_dynamic}, we impose $\\alpha_i^s + \\beta_i^s<1$ for all $i=1,2,\\dots,N$ and $s=1,2,\\dots,L$. Moreover, we need to ensure that $\\mathbf{C}_{t}^s$ is positive defined matrix for each $t=1,2,\\dots,T$ and across all the possible states of nature $s=1,2,\\dots,L$.\nThe only unspecified quantity in equation \\eqref{eq:corr_dynamic}, is the forcing variable $\\bXi_t$ which is set to the moving average variance--covariance matrix of length $m$\n\\begin{equation}\n\\bXi_t = m^{-1}\\sum_{j=0}^m \\mathbf{u}_{t-m+j}\\mathbf{u}_{t-m+j}^\\prime - \\bar\\mathbf{u}_{m,j}\\bar\\mathbf{u}_{m,j}^\\prime,\n\\end{equation}\nwhere $\\bar\\mathbf{u}_{m,j} = m^{-1}\\sum_{j=0}^m \\mathbf{u}_{t-m+j}$ represents the vector containing the simple mean of the pseudo observations across the period $\\left\\{t-m;t\\right\\}$. The proposed model is similar those of \\cite{vargas.2008} and \\cite{chou_liao.2008} but differs for the chosen forcing variable $\\bXi_t$ which updates the conditional variance--covariance matrix. The proposed updating scheme makes use of a\nrolling window variance--covariance matrix as forcing variable by which we can get a smooth time evolution of the correlation.\nAs regards this aspect, the proposed correlation dynamics generalises the approach of \\cite{patton.2006b} and \\cite{jondeau_rockinger.2006b} in a fully multivariate setting.\\newline\n\\indent More interesting, the model here proposed, allows for the dynamic of the variance--covariance matrix to be also subject to the realisation of a first order Markov process. Using a different approach, \\cite{bernardi_catania.2015a} show that including this regime dependent behaviour into the variance--covariance dynamics, really helps in describing the time--varying dependence of equity indexes. Indeed, ad documented, for example, by \\cite{pelletier.2006}, \\cite{guidolin_timmermann.2008}, \\cite{ang_bekaert.2002}, \\cite{bernardi_catania.2015a}, there is strong evidence of the presence of different regimes affecting the dependence structure of equity assets. Obviously, the model proposed in equation \\eqref{eq:corr_dynamic} can be easily generalised to account for the leverage effect in a similar way as for the AGDCC of \\cite{cappiello_etal.2006}\n\\begin{align}\n\\mathbf{C}_{t+1}^s =&\\left(\\bar\\mathbf{C}-\\mathbf{A}^s\\bar\\mathbf{C}{\\mathbf{A}^{s}}^\\prime-\\mathbf{B}^s\\bar\\mathbf{C}{\\mathbf{B}^{s}}^\\prime-\\mbox{\\boldmath $\\Gamma$}^s\\mathrm{\\bar\\mathbf{N}}^s{\\mbox{\\boldmath $\\Gamma$}^{s}}^\\prime-\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\bar\\mathbf{X} \\right)\\nonumber\\\\\n&\\qquad\\qquad\\qquad+\\mathbf{A}^s\\bXi_t{\\mathbf{A}^{s}}^\\prime+\\mathbf{B}^s\\mathbf{C}_t^s{\\mathbf{B}^{s}}^\\prime+\\mbox{\\boldmath $\\Gamma$}^s\\boldsymbol\\eta_t^s{\\boldsymbol\\eta_t^s}^\\prime{\\mbox{\\boldmath $\\Gamma$}^{s}}^\\prime+\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\mathbf{X}_t,\n\\label{eq:Gdynamic}\n\\end{align}\nwhere $\\mbox{\\boldmath $\\Gamma$}^s=\\left\\{\\sqrt{\\gamma_i^s}\\right\\}$, $i=1,2,\\dots,N$ is a diagonal matrix, and $\\boldsymbol\\eta_t^s$ is a vector whose $i$--th component is given by $\\bbone\\left(x_{i,t}^s<0\\right)$ where $x_{i,t}^s=\\mathcal{T}_{\\nu_\\mathsf{c}^s}^{-1}\\left(u_{it}\\right)$ for $i=1,2,\\dots,N$, and $\\mathcal{T}_{\\nu_\\mathsf{c}^s}^{-1}\\left(\\cdot\\right)$ denotes the inverse cumulative density function (cdf) of a standardised Student--t distribution with $\\nu_\\mathsf{c}^s$ degree of freedom, and $\\bbone\\left(\\cdot\\right)$ represents the indicator function.\nIn equation \\eqref{eq:Gdynamic}, $\\mathrm{\\bar\\mathbf{N}}^s$ is the unconditional empirical average of the matrices $\\boldsymbol\\eta_t^s{\\boldsymbol\\eta_t^s}^\\prime$, $t=1,2,\\dots,T$. Several constraints must be imposed in order to avoid explosive patterns in the conditional variance--covariance matrix dynamic in equation \\eqref{eq:Gdynamic}, see, e.g., \\cite{cappiello_etal.2006}. Since in our empirical investigation we do not find any substantial improvement in favour of model \\eqref{eq:Gdynamic} with respect to model \\eqref{eq:corr_dynamic}, throughout the paper we will continue to refer to the specification without leverage effect. Moreover, since the number of parameters in the model in described in equation \\eqref{eq:corr_dynamic} grows linearly with respect to the number of asset $N$, sometimes it would be convenient to set $\\alpha_1^s = \\alpha_2^s = \\dots = \\alpha_N^s=\\alpha^s$ and $\\beta_1^s = \\beta_2^s = \\dots = \\beta_N^s=\\beta^s$. We name this constrained specification `` simple'', while we refer to the aforementioned general case with the name `` generalised''.\\newline\n\\indent To conclude the model specification, we report the multivariate density function of the random vector $\\mathbf{y}_t=\\left(y_{1,t},y_{2,t},\\dots,y_{N,t}\\right)^\\prime$ which is given by\n\\begin{align}\n\\mathbf{y}_{t+1}\\mid\\mathbf{y}_{t}\\sim h\\left(\\mathbf{y}_{t+1};\\mathbf{\\theta}_{t+1},\\mathbf{X}_{t},\\mbox{\\boldmath $\\Delta$}_{t+1}^s, s=1,\\dots,L\\right),\n\\label{eq:multivariate_density}\n\\end{align}\nwhere $\\mathbf{\\theta}_t=\\left(\\mathbf{\\theta}_{1,t}^\\prime,\\mathbf{\\theta}_{2,t}^\\prime,\\dots,\\mathbf{\\theta}_{N,t}^\\prime\\right)^\\prime$, and\n\\begin{align}\nh\\left(\\mathbf{y}_{y+1};\\cdot\\right)=&\\prod_{i=1}^N f_{\\mathcal{AST}}\\left(y_{i,t+1};\\mathbf{\\theta}_{i,t+1}\\right)\n\\sum_{s=1}^L\\pi_{t+1\\vert t}^{(s)} c_T\\left(F_{\\mathcal{AST}}\\left(\\mathbf{y}_{t+1};\\mathbf{\\theta}_{i,t+1}\\right),\\mbox{\\boldmath $\\Delta$}_{t+1}^s,\\mathbf{X}_t\\right)\\nonumber,\n\\end{align}\nwith $\\mbox{\\boldmath $\\Delta$}_{t+1}^s=\\left(\\mathbf{R}_{t+1}^s,\\nu_{\\mathsf{c}}^s\\right)$ and the mixing weight $\\pi_{t+1\\vert t}^{(s)}$ are\n\\begin{equation}\n\\pi_{t+1\\vert T}^{(s)}=\\sum_{m=1}^L {q}_{m,s}\\mathbb{P}\\left(S_{t+1}=m\\mid\\mathbf{y}_{1:t},\\mathbf{X}_{1:t}\\right),\\nonumber\n\\end{equation}\nfor $s=1,2,\\dots,L$, and $\\mathbb{P}\\left(S_{t+1}=m\\mid\\mathbf{r}_{1:t},\\mathbf{X}_{1:t}\\right)$ can be evaluated using the well known FFBS algorithm detailed in \\cite{fruhwirth_schnatter.2006}. Here, the $i$--th marginal density and probability functions are represented by $f_\\mathcal{AST}\\left(\\cdot\\right)$ and $F_\\mathcal{AST}\\left(\\cdot\\right)$ while the copula probability density is denoted by $c_T\\left(\\cdot\\right)$.\\newline\n\\indent Concerning the estimation of the proposed model, unfortunately, given the presence of the exogenous covariates, we cannot rely on the usual results for DCC models such as, for examples, those of \\cite{engle.2002} and \\cite{cappiello_etal.2006}, while, the Constrained ML (CML) approach used by \\cite{vargas.2008} and \\cite{chou_liao.2008} in a similar context, requires the evaluation of highly nonlinear constraint during the maximisation procedure. We follow a different approach which consists on penalising the log--likelihood function (PML) in the second step of the IFM procedure, whenever the variance--covariance matrix is not positive defined. However, this solution is not optimal since it introduces strongly nonlinearities in the likelihood shape, and it may result in possible local optima solutions of the maximisation problem. Hence, good starting values are required.\nIn the second step of the IFM procedure, we made use of the Expectation--Maximisation algorithm of \\cite{dempster_etal.1977} in order to deal with the markovian structure characterising the latent states. Further details about the employed estimation technique can be found in \\cite{bernardi_catania.2015a}.\n\\section{Asset Allocation Strategy}\n\\label{sec:asset_allocation}\n\\noindent In this section, we tailor the `` distribution timing\\qmcsp approach to the portfolio optimisation problem of \\cite{jondeau_rockinger.2012} to the dynamic MS copula approach introduced in the previous Section. Specifically, we consider a rational investor having full information about the whole distribution of his\/her future wealth and builds a dynamic asset allocation strategy by maximising his\/her expected utility. It is worth noting that, higher moments play a fundamental role during the utility maximisation process.\nFormally, let $y_{p,t+1} = \\sum_{i=1}^N\\lambda_{i,t\\vert t+1} y_{i,t+1}$ be the future portfolio return, where $\\lambda_{i,t\\vert t+1}$ denotes the portion of today wealth that the investor is willing to allocate in the $i$--th risky asset, with $\\sum_{i=1}^N \\lambda_{i,t\\vert t+1}=1$, then $W_{t+1} = 1 + y_{p,t+1}$ represents the future portfolio wealth at time $t+1$\\footnote{We will always consider an initial wealth of 1\\$. This choice will not affect the portfolio decision given our assumption on the investor's utility function.}. The vectors, $\\mathbf{y}_{t+1} = \\left(y_{1,t+1},y_{2,t+1},\\dots,y_{N,t+1}\\right)^\\prime$ and $\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}=\\left(\\lambda_{1,t\\vert t+1},\\lambda_{2,t\\vert t+1},\\dots,\\lambda_{N,t\\vert t+1}\\right)^\\prime$ contain all the assets in which the investor can open a speculative position at time $t$, which is subsequently held for the period $\\left[t,t+1\\right)$ and the portfolio weights, respectively. The portfolio optimisation problem can then be written as follows\n\\begin{equation}\n\\arg\\max_{\\left\\{\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right\\}}\\mathbb{E}_t\\left[\\mathcal{U}\\left(1 + \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime \\mathbf{y}_{t+1}\\right)\\right],\\nonumber\n\\label{eq:maximisation}\n\\end{equation}\nwhere $\\mathcal{U}\\left(W_{t+1}\\right)$ denotes the investor utility function and the expectation must be taken with respect to the future portfolio return distribution. Clearly, this problem does not have a closed--form solution given that $\\mathbf{y}_{t+1}$ is conditionally distributed according to equation \\eqref{eq:multivariate_density}. Moreover, when $N$ is large, numerical techniques such as, for example, Monte Carlo integration and quadrature rules become infeasible. Nevertheless, the maximisation problem can be easily solved by taking a Taylor expansion of the utility function around the wealth at time $t$, $W_t$, i.e., by considering\n\\begin{equation}\n\\mathcal{U}\\left(W_{t+1}\\right) = \\sum_{k=0}^\\infty\\frac{\\mathcal{U}^{\\left(k\\right)}\\left(W_t\\right)}{k!}\\left(W_{t+1} - W_t\\right)^k,\n\\label{eq:taylor_expansion}\n\\end{equation}\nwhere $\\mathcal{U}^{\\left(k\\right)}$ denotes the $k$--th derivative of the utility function with respect to its argument. Taking the expectation at time $t$ of both sides of equation \\eqref{eq:taylor_expansion} we get\n\\begin{equation}\n\\mathbb{E}_t\\left[\\mathcal{U}\\left(W_{t+1}\\right)\\right] =\\sum_{k=0}^\\infty\\frac{\\mathcal{U}^{\\left(k\\right)}\\left(W_t\\right)}{k!}m_{p,t+1}^{\\left(k\\right)},\n\\label{eq:taylor_expansion_portfolio}\n\\end{equation}\nwhere $m_{p,t+1}^{\\left(k\\right)}=\\mathbb{E}\\left[r_{p,t+1}^k\\right]=\\mathbb{E}\\left[\\left(W_{t+1}-W_t\\right)^k\\right]$ denotes the $k$--th non central moment of the portfolio return distribution at time $t+1$. If we consider a fourth--order Taylor expansion, i.e., we truncate equation \\eqref{eq:taylor_expansion_portfolio} at $\\bar{k}=4$, the first four central moments of the portfolio distribution are immediately available as a function of the corresponding non--central moments and the portfolio weights. More precisely, the centred moments of the portfolio returns, at time $t+1$, are\n\\begin{align}\n\\mu_{p,t+1} &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{1,t+1}\\nonumber\\\\\n\\sigma_{p,t+1}^2 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{2,t+1}\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\nonumber\\\\\ns_{p,t+1}^3 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{3,t+1}\\left(\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right)\\nonumber\\\\\nk_{p,t+1}^4 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{4,t+1}\\left(\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right),\\nonumber\n\\end{align}\nwhere $\\otimes$ stands for the Kronecker product, and the corresponding non--centred moments required by equation \\eqref{eq:taylor_expansion_portfolio} can be obtained as\n\\begin{align}\nm_{p,t+1}^{\\left(1\\right)} &= \\mu_{p,t+1} \\nonumber\\\\\nm_{p,t+1}^{\\left(2\\right)} &= \\sigma_{p,t+1}^2 + \\mu_{p,t+1}^2 \\nonumber\\\\\nm_{p,t+1}^{\\left(3\\right)} &= s_{p,t+1}^3 + 3\\sigma_{p,t+1}^2\\mu_{p,t+1} + \\mu_{p,t+1}^3\\nonumber\\\\\nm_{p,t+1}^{\\left(4\\right)} &= k_{p,t+1}^4 + 4s_{p,t+1}^3\\mu_{p,t+1} + 6\\sigma_{p,t+1}^2\\mu_{p,t+1}^2 + \\mu_{p,t+1}^4.\\nonumber\n\\end{align}\nNote that $M_{1,t+1}$, $M_{2,t+1}$, $M_{3,t+1}$, $M_{4,t+1}$ are the mean, the variance--covariance matrix, the co--skewness and co--kurtosis of the conditional joint density function in equation \\eqref{eq:multivariate_density}, respectively. Here, the original $\\left(N,N,N\\right)$ and $\\left(N,N,N,N\\right)$ matrices of co--skewness and co--kurtosis were transformed into the $\\left(N,N^2\\right)$ and $\\left(N,N^3\\right)$ matrices, respectively, as suggested by \\cite{deathayde_flores.2004} by slicing each $\\left(N,N\\right)$ layer and pasting them, in the same order, sideways.\\newline\n\\indent Having defined the general portfolio optimisation framework, the only quantity that remains unspecified is the specific utility function the rational investor maximises. We assume that our agent makes his\/her investment choices according to a Constant Relative Risk Aversion (CRRA) utility function defined as\n\\begin{equation}\n\\mathcal{U}_{t+1}\\left(W_{t+1}\\right) = \\begin{cases} \\frac{W_{t+1}^{1-\\upsilon}}{1-\\upsilon}, & \\mbox{if } \\upsilon>1 \\\\ \\log\\left(W_{t+1}\\right), & \\mbox{if } \\upsilon=1, \\end{cases}\n\\label{eq:utility}\n\\end{equation}\nwhere $\\upsilon$ is the coefficient of relative risk aversion, which is assumed to be constant with respect to the wealth $W_{t+1}$. For CRRA utility function, equation \\eqref{eq:taylor_expansion_portfolio} truncated at the fourth order becomes\n\\begin{align}\n\\mathbb{E}_t\\left[\\mathcal{U}\\left(W_{t+1}\\right)\\right]&\\approx\\frac{1}{1-\\upsilon} + m_{p,t+1}^{\\left(1\\right)} - \\frac{\\upsilon}{2} m_{p,t+1}^{\\left(2\\right)}\n+\\frac{\\upsilon\\left(\\upsilon + 1\\right)}{6} m_{p,t+1}^{\\left(3\\right)} -\n\\frac{\\upsilon\\left(\\upsilon + 1\\right)\\left(\\upsilon + 2\\right)}{24} m_{p,t+1}^{\\left(4\\right)},\n\\label{eq:utility_final}\n\\end{align}\nwhere the approximation order is $O\\left(5\\right)$. The closed--form expression for the expected utility makes the maximisation of equation \\eqref{eq:utility_final} with respect to the portfolio weights, straightforward. However, unfortunately, there is no closed--form solution for the moments of order two to four of the multivariate joint density function and we need to resort to numerical procedures. \\ref{sec:appendix_Moments} describes the method used to approximate the moments of the predictive joint density.\n\\section{Empirical study}\n\\label{sec:empirical_study}\n\\noindent In this section we apply the model and the portfolio optimisation methodology introduced in previous sections to deliver a set of optimal portfolio weights. To this end, we consider the same data set of \\cite{jondeau_rockinger.2006a} which consists of five of the major international equity indexes, namely the Standard \\& Poors 500 (SPX), the Nikkei 225 (N225), the FTSE 100 (FTSE), the DAX30 (GDAXI), and the CAC40 (FCHI). At each time $t$ over the out of sample period, the rational investor holding the portfolio is expected to choose the optimal allocation of the wealth by maximising his\/her expected utility using the argument presented in Section \\ref{sec:asset_allocation}. The illustration of the empirical results firstly focus on the marginal distribution, then deals with the multivariate density estimation and forecast and concludes with the presentation of the optimal portfolio allocations.\n\\subsection{Data}\n\\label{sec:data}\n\\noindent The data set consists of 1449 weekly returns spanning the period from the 8--th January, 1988, to the 9--th October, 2015. Starting from this series, the last 449 observations spanning the period from the 9--th March, 2007 to the end of the sample, are used to perform the out of sample analysis and to backtest the portfolio strategy, while the fist 1000 observation are used to fit the model and assess its in--sample performance. Table \\ref{tab:Index_data_summary_stat} reports descriptive statistics of the indexes returns over the in--sample and out--of--sample periods. As expected, we find strong evidence of departure from normality, since all the considered series appear to be leptokurtic and skewed. Moreover, the \\cite{jarque_bera.1980} statistic test strongly rejects the null hypothesis of normality for all the considered series at the 1\\% confidence level. It is worth noting that, the departure from normality, appears to be stronger during the out--of--sample period. As discussed for example by \\cite{shiller_2012}, this empirical evidence can be considered as an effect of the recent GFC of 2007--2008 that affected the overall economy. Table \\ref{tab:Dependence} reports the linear correlation and the Kendall $\\tau$ unconditional estimates. All the coefficients seem to be substantially different between the two subperiods, suggesting a time varying behaviour of the dependence structure that characterise our series. To further investigate this aspect, we perform the LMC test for constant versus time--varying conditional correlation of \\cite{tse.2000} on the returns belonging to the whole period. In our case the LMC test is distributed according to a $\\chi_{10}^2$ with a critical value of about 23.21 at the 1\\% confidence level, while the test statistic is about 997 which strongly adverses the null hypothesis of time invariant correlation. Moreover, the lower triangular parts of Table \\ref{tab:Dependence} also gives an insights about the changes in the tail of the joint probability distribution. These findings strongly support for the Student--t copula DCC model with exogenous regressors introduced in the previous sections \\ref{sec:marginal_model}--\\ref{sec:multivariate_model}, being able to describe the evolution of the whole joint distribution of assets returns.\\newline\n\\indent As previously discussed, one of the main feature of the proposed model, concerns its ability to easily incorporate exogenous information in the state dependent correlation dynamics. Consequently, to control for the general economic conditions, we use observations of the following macroeconomic regressors as suggested by \\cite{adrian_brunnermeier.2011}, \\cite{chao_etal.2012} and \\cite{bernardi_etal.2015}:\n\\begin{enumerate}\n\\item[(I)] the weekly change in the three--month Treasury Bill rate (3MTB);\n\\item[(II)] the weekly change in the slope of the yield curve (TERMSPR), measured by the difference of the 10-year Treasury rate and the 3--month Treasury Bill rate;\n\\item[(III)] the weekly change in the credit spread (CREDSPR) between 10--year BAA rated bonds and the 10-year Treasury rate.\n\\end{enumerate}\nHistorical data for all the considered exogenous regressors are freely available on the Federal Reserve Bank of St. Louis (FRED) web site.\n\\subsection{Parameters estimation and goodness--of--fit results}\n\\label{sec:parameters_est}\n\\noindent The empirical experiment is conducted as follows. First, we estimate the marginals and the MS copula DCC models over the in--sample period, and then we make a one--step ahead rolling forecast of the multivariate joint density of the returns for the entire out of sample period. During the rolling forecast exercise, the rational investor takes an investment decision in terms of wealth allocation, at each point in time $t+s$ conditional to the information set available at time $t+s-1$, for $s=1,2,\\dots,448$. For both the marginal and copula models we re--estimate the model parameters each 24 observations, corresponding to six months using a fixed moving window.\\newline\n\\indent Table \\ref{tab:Marginal_estimates} reports the estimated coefficients for each GAS--AST marginal model. We find the scores coefficients associated with the scale $\\left(\\sigma\\right)$ and shape $\\left(\\nu\\right)$ parameters strongly significant indicating that our marginal model is able to capture the changes in the volatility patterns as well as in the tails of the conditional marginal distributions. On the contrary, the scores coefficients associated with the location $\\left(\\mu\\right)$ and the skewness $\\left(\\gamma\\right)$ parameters of the conditional AST distribution are not statistically different from zero. Moreover, we find that the shape and scale parameters of the AST distribution display high persistence, while the location and skewness parameters do not display such behaviour. The coefficients related to the scaled scores have quite similar magnitude across all the assets, indicating that the GAS updating mechanism adapts quite well to the different characteristics of the returns.\\newline\n\\indent Before moving to the copula specification, we asses the goodness of fit of the marginal models. In particular, we want to test if the PITs of the estimated marginal densities are independently and identically distributed uniformly on the unit interval. To this end, we perform the same test employed by \\cite{vlaar_palm.1993}, \\cite{jondeau_rockinger.2006b}, and \\cite{tay_etal.1998}. The test of the iid Uniform$\\left(0,1\\right)$ assumption consists of two parts.\nThe first part concerns the independent assumption, and it tests if all the conditional moments of the data, up to the fourth one, have been accounted for by the model, while the second part checks if the AST assumption is reliable by testing if the PITs are Uniform over the inverval $\\left(0,1\\right)$. In order to test if the PITs are independently and identically distributed, we define $\\hat u_{i,t}=F_\\mathcal{AST}\\left(y_{i,t},\\hat\\mathbf{\\theta}_{i,t}\\right)$ as the PIT of return $i$ at time $t$ and $\\bar{u}_i = T^{-1}\\sum_{t=1}^T \\hat u_{i,t}$ as the empirical average of the $i$--th PIT series, then we examine the serial correlation of $\\left(\\hat u_{i,t} - \\bar{u}_i\\right)^k$ for $k=1,2,\\dots,4$ by regressing each $\\left(\\hat u_{i,t} - \\bar{u}_i\\right)$ for $i=1,2,\\dots,N$ on its own lags up to the order 20.\nThe hypothesis of no serial correlation can be tested using a Lagrange multiplier test defined by the statistics $\\left(T-20\\right)R^2$ where $R^2$ is the coefficient of determination of the regression. This test is distributed according to a $\\chi^2\\left(20\\right)$ with a critic value of about 31.4 at the 5\\% confidence level. The first four columns of Table \\ref{tab:Uniform_test}, named ${\\rm DGT}-{\\rm AR}^{\\left(k\\right)}$, report the test for $k=1,2,3,4$. Our results confirm that, for almost all the considered time series and all the moments of the univariate conditional distributions, the marginal models are able to effectively account for the dynamic behaviour of the series. For what concerns the Uniform assumption of the PITs, we again employ the test suggested by \\cite{tay_etal.1998} which consists in splitting the empirical distribution of $\\hat u_{i,t}$ into $G$ bins and test whether the empirical and the theoretical distributions significantly differ on each bin. More precisely, let us define $n_{q,i}$ as the number of $\\hat u_{i,t}$ belonging to the bin $q$, it can be shown that\n\\begin{equation}\n\\xi_i=\\frac{\\sum_{q=1}^G\\left(n_i-\\mathbb{E} n_i\\right)^2}{\\mathbb{E} n_i} \\sim \\chi ^2\\left(G-1\\right),\n\\end{equation}\nwhere $G$ is the number of bins. In our case, since we have estimated the model parameters, the asymptotic distribution of $\\xi_i$ is bracketed between a $\\chi ^2\\left(G-1\\right)$ and $\\chi ^2\\left(G-m-1\\right)$ where $m$ is the number of estimated parameters.\nIn order to be consistent with the results reported by \\cite{tay_etal.1998} and \\cite{jondeau_rockinger.2006b}, we choose $G=20$ bins using a $\\chi^2\\left(19\\right)$ distribution for $\\xi_i,\\quad i=1,2,\\dots, N$, see also \\cite{vlaar_palm.1993}. The test statistic of the Uniform test are reported in the last column of Table \\ref{tab:Uniform_test} and are named $\\mathrm{DGT-H}\\left(20\\right)$.\nWe can observe that, for each series, the PITs are uniformly distributed over the interval $(0,1)$ at a confidence level lower then $1\\%$.\nThese findings definitely confirm the adequacy of our assumption on the innovation term and the conditional returns dynamic.\\newline\n\\indent We now move to study the dependence structure of the series using the proposed copula model. As previously discussed, various parameterisations of the proposed model for the conditional dependence structure of assets returns can be employed. Given the available data set, several different levels of flexibility are possible. Model specification concerns the selection of either the number of states, or the `` simple'', or `` generalised'', specifications of the joint model discussed in Section \\ref{sec:multivariate_model}, or the inclusion of the information coming from the exogenous data.\nIn particular, the choice of the number of regimes, in the HMM literature is an open question. As argued by \\cite{lindsay.1983} and \\cite{bohning.2000}, the number of states can be either estimated or tested. However given the large number of model combinations we are considering here, we decide to follow the most common practice of choosing the number of states according to the BIC, see e.g. \\cite{cappe_etal.2005}, \\cite{fruhwirth_schnatter.2006} and \\cite{zucchini_macdonald.2009}. We chose the best model according to the BIC since we want to penalise more the models with a high number of parameters. Moreover, since the marginal models are invariant to the choice of the possible dependence specifications, we report the log--likelihood and the information criteria only for the copula specifications.\nTable \\ref{tab:Model_Choice} reports the log--likelihood evaluated at its optimum, the BIC and the AIC for different combinations of number of states, type of dynamic and inclusion of exogenous covariates. According to the BIC, the best model is the '' simple\\qmcsp DCC dynamic specification with two regimes and the inclusion of the exogenous covariates. The BIC clearly select the model with the highest ability to represent different levels of the dependence structure affecting the data, and to react on the exogenous information depending on the state of the world. Table \\ref{tab:CopulaCoef} reports the estimated coefficients for the best fitting model.\nWe can observe that both regimes are characterised by strong persistence in the dynamic of the conditional dependence, however the dynamics react differently to the new endogenous and exogenous information. More precisely, we can observe that in the second regime, the coefficient $\\mathbf{A}^s$, $s=2$, which measures the reaction to new endogenous information, is much higher than the same coefficient in the other regime. This means that the second regime reacts more quickly, compared to the first one, when a change occurs in the market dependence structure. On the contrary, we observe a similar persistent behaviour of the correlation matrix within the two regimes. For what concerns the estimated state specific degree of freedom, we note that the second regime is characterised by a smaller coefficient indicating higher positive as well as negative tail dependence compared to the second one. These finding is also consistent with the empirical evidence reported by \\cite{chollete_etal.2009}, \\cite{bernardi_etal.2015} and \\cite{bernardi_catania.2015a}, in related studies.\nLooking at the estimated coefficients associated to the exogenous covariates, we can observe different behaviours of the conditional correlation matrix, depending on the regimes. In particular, we note that the returns dependence, conditional of being in the first regime, appears to react less to an absolute increase of the considered exogenous covariates. On the contrary, the dependence structure conditional of being in the second regime, displays a stronger reaction to the macroeconomic factors here considered.\nThe estimated transition probabilities of the unobserved Markov chain are $p_{11}=0.984$ and $p_{22}=0.987$, indicating quite strong persistence and an expected duration of the first and second states of about 61.5 and 75.9 weeks, respectively. Finally, we found that the dependence behaviour of the considered US market returns switched according to the 2000--2003 dot--com bubble and the recent 2008--2010 Global Financial Crisis.\n\\subsection{Portfolio allocation}\n\\noindent Now we consider the portfolio allocation problem. As stated in the Introduction, we focus on a rational investor who maximise his\/her expected utility. Specifically, once the models parameters have been estimated over the in sample period, the investor forecasts the one--step ahead joint distribution of his\/her assets return for each time $t$ of the out--of--sample period, and then he\/she allocates the available wealth among those assets accordingly to the portfolio allocation procedure discussed in Section \\ref{sec:asset_allocation}. We assume an initial wealth of 1\\textdollar, an allocation strategy that allows for either long and short positions and a risk free rate equal to zero. The choice of the level of the initial wealth is otherwise irrelevant because it does not influence the risk aversion, and hence, the allocation strategy under the defined CRRA utility function reported in equation \\eqref{eq:utility}. Moreover, in order to evaluate the influence of the level of risk aversion in the portfolio allocation strategy, we consider different levels of relative risk aversion, i.e., $\\upsilon=3,7,10,20$. In order to asses the performance of the proposed model from a portfolio optimisation viewpoint, we run an horse race with five common alternatives:\n\\begin{itemize}\n\\item[-] the `` Equally Weighted portfolio'', (EW) strategy, where the weights associated to each asset are kept fixed and equal to $1\/N$ during the validation period;\n\\item[-] the `` Minimum Variance portfolio'', (MV) strategy, where the weights associated to each asset are kept fixed and equal to those minimising the portfolio variance during the in--sample period;\n\\item[-] the `` DCC\\qmcsp strategy, where the one--step ahead forecast of the conditional mean and the conditional variance--covariance matrix of the joint distribution of assets return are performed using the DCC$\\left(1,1\\right)$ model of \\cite{engle.2002}, whose parameters are estimated by Quasi--ML, see, e.g., \\cite{francq_zakoian.2011}. The weights of each asset are then chosen by maximising equation \\eqref{eq:maximisation} truncated at the second order;\n\\item[-] the `` Naive Mean--Variance'', (NMV) and `` Naive Higher--Moments'', (NHM) strategies, where the optimal weights are chosen again by maximising equation \\eqref{eq:maximisation}, as in the case of the `` DCC\\qmcsp strategy, and the first four moments of the joint distribution of assets return are estimated using a rolling window approach.\n\\end{itemize}\nHereafter, we label the optimal investment strategy according to the model detailed in Section \\ref{sec:model}, Flexible Dynamic Dependence Model (FDDM).\\newline\n\\indent One easy way to compare different portfolios performance in terms of realised utility is to consider the so called `` management fee\\qmcsp approach. The management fee approach relies on the definition of the amount of money, $\\vartheta$, a rational investor is willing to pay, or gain, in order to switch from the portfolio that he\/she is currently holding on a given period, to a different one, and it coincides with the solution of the the following equation\n\\begin{equation}\nS^{-1}\\sum_{t=F+1}^{F+S}\\mathcal{U}\\left(1+\\mbox{\\boldmath $\\lambda$}^{\\mathcal{A}\\prime}_{t\\vert t+1}\\mathbf{y}_{t+1}\\right) = S^{-1}\\sum_{t=F+1}^{F+S}\\mathcal{U}\\left(1+\\mbox{\\boldmath $\\lambda$}^{\\mathcal{B}\\prime}_{t\\vert t+1}\\mathbf{y}_{t+1}-\\vartheta\\right),\n\\label{eq:managementFee}\n\\end{equation}\nwhere $F$ and $S$ are the length of the in--sample and out--of--sample periods, respectively, and, as before, $\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}$ denotes the vector of amounts of wealth $\\lambda_{j,t\\vert t+1}$, the investor is allocating to each asset $j=1,2,\\dots,N$ during the period $\\left(t,t+1\\right]$. From equation \\eqref{eq:managementFee}, it is easy to see that if $\\vartheta>0$, the investor is willing to pay a positive amount of money in order to switch from portfolio $\\mathcal{A}$ to portfolio $\\mathcal{B}$. On the contrary, if $\\vartheta<0$, the investor is going to ask for a higher return from portfolio $\\mathcal{B}$ in order to compensate the loss of utility he\/she would experience for switching from $\\mathcal{A}$ to $\\mathcal{B}$. Finally, if $\\vartheta=0$ the two portfolios give exactly the same utility to the investor, leaving he\/she indifferent between the two options. Table \\ref{tab:MF_mSR} reports the evaluated management fees for all the possible couples made by he FDDM strategy and a competitor, under different values of relative risk aversion coefficient $\\upsilon=\\left(3,5,10,20\\right)$. More precisely, looking at equation \\eqref{eq:managementFee}, we fix $\\mathcal{B}$ equal to the FDDM strategy, and we let $\\mathcal{A}$ to vary among the competing alternatives, i.e., the strategy a $\\mathcal{A}$ is selected among the alternatives $\\{\\mathrm{EW},\\mathrm{MV},\\mathrm{DCC},\\mathrm{NMV},\\mathrm{NHM}\\}$.\nAs a consequence, if the reported management fee is positive, then the optimal portfolio based on the proposed model is better compared to the specific competitor, otherwise the alternative strategy is preferred. Another useful indicator to compare portfolio strategies is the Sharpe ratio. However, since this indicator does not result in a measure of outperformance over alternative strategies with different levels of risk, we employ the so called modified Sharpe Ratio (mSR) introduced by \\cite{graham_harvey.1997} which is defined as\n\\begin{equation}\n\\mathrm{mSR}\\left(\\mathcal{A},\\mathcal{B}\\right)=\\frac{\\sigma_{\\mathcal{A}}}{\\sigma_\\mathcal{B}}\\mu_\\mathcal{B} - \\mu_\\mathcal{A},\n\\label{eq:modSharpeRatio}\n\\end{equation}\nwhere it immediately follows that if $\\mathrm{mSR}\\left(\\mathcal{A},\\mathcal{B}\\right)>0$, then model $\\mathcal{B}$ is preferred to model $\\mathcal{A}$ under a simple mean--variance preference ordering.\\newline\n\\indent Table \\ref{tab:MF_mSR} reports the management fee and the mSR for all the considered alternatives against the FDMM strategy. We note that, almost all the management fees and the Sharpe ratios are positive and significantly different from zero, indicating economic value in favour of the FDDM strategy. Indeed, concerning the evaluated management fees, we found that the FDDM strategy is almost always preferred by a rational investors with arbitrary level of risk aversion. The only exceptions are when the FDDM strategy is compared with the DCC strategy assuming an high degree of risk aversion $\\left(\\upsilon=3\\right)$ and with the EW strategy assuming a low degree of risk aversion $\\left(\\upsilon=20\\right)$. The results associated with the modified Sharpe ratio are in line with those reported by the management fees analysis. The only difference is that, when the degree of risk aversion is low, $\\left(\\upsilon=20\\right)$, the statistical significance of the modified Sharpe ratio vanishes, indicating that there is no evidence of outperformance between the FDDM and the alternative strategies.\nTable \\ref{tab:avg_weights} reports the average portfolio weights over the whole out of sample period associated with each strategy. We note that the positions taken by the FDDM and DCC strategies are quite homogeneous in terms of which asset to buy and which asset to sell. Indeed, the N225 and the FCHI indexes are always shorted by the FDDM and DCC strategies. These finding suggest that the signal coming from the two strategies is similar, demonstrating that a truly dynamic model is important when investment positions are concerned. We also note that, the weights associated with the NMV, NHM, and MVP are very similar, independently from the assumed degree of risk aversion of the investor. Looking at the descriptive statistics of the indexes given in Table \\ref{tab:Index_data_summary_stat} we can observe that the assets with the higher (in absolute value) wealth allocation, according to the FDDM strategy, are those who, ex--post, result in lower levels of kurtosis. On the contrary, the indexes with the lower exposition, report high kurtosis during the out--of--sample period. More interesting, looking again at Table \\ref{tab:Index_data_summary_stat}, we note that, during the in--sample period, the levels of kurtosis among assets were quite homogeneous, which means that our model is able to effectively forecast the changes in the behaviour of the joint density.\nMore precisely, on the one hand, the GAS updating mechanism is suited to effectively move the parameters through the direction of higher probability mass, and, on the other hand, the Markov switching dynamic copula model helps in capturing the evolution of the dependence structure during time and across states.\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\noindent In this paper we propose a new flexible copula model being able to account for a wide variety of stylised facts affecting multivariate financial time series. Specifically, we allow the dependence structure affecting the considered financial time series to depend either on the realisation of a first order Markov chain and on exogenous covariates that may represent economic factors such as, interest rates, GDP growth rates, as well as financial indexes such as realised volatility indexes. The proposed solution is highly flexible, since the dependence dynamics account for either observable and unobservable latent factors.\nAnother interesting contribution concerns the marginal model specification. We develop a novel GAS filter \\citep{creal_etal.2013, harvey.2013} based on the Asymmetric Student--t distribution of \\cite{zhu_galbraith.2010} where all the location, scale and shape parameters evolves according to a first order transition dynamic based on the scaled score of the conditional distribution. In this way we allow for time variation in the first four moments of the conditional marginal distribution.\nParameters estimation is performed by exploiting the two step Inference Function for Margins procedure detailed in \\cite{patton.2006a} for conditional copulas, where in the second step we made use of the Expectation--Maximisation algorithm of \\cite{dempster_etal.1977}. Further details about the employed estimation technique can be found in \\cite{bernardi_catania.2015a}.\\newline\n\\indent The proposed model is applied to predict the evolution of the weekly returns of five major international stock indexes, previously considered by \\cite{jondeau_rockinger.2006a}, over the period 1988--2015. In the empirical application, in sample and out of sample performances of the proposed model are deeply investigated. The in sample analysis, reveals statistical evidence for nonlinearities and highlights the relevance of the inclusion of exogenous covariates in explaining the dependence structure affecting the indexes returns. Furthermore, our analysis confirms that our model can be effectively used to describe and predict various aspects of the conditional joint distribution of equity returns, such as time varying means, variances, co--skewness and co--kurtosis. Out of sample model performances are evaluated by solving an optimisation problem where a rational investor with power utility function takes future investment positions according to the predictive density of assets returns.\nSpecifically, we compare the optimal portfolio strategy implied by the proposed model with that of several static and dynamic alternatives. The portfolio optimisation strategy is based on the Taylor expansion of a CRRA utility function of the rational investor and involves the evaluation of the portfolio moments up to the fourth order. Since higher order moments of the joint distribution are not analytically known we resort to numerical integration. Our empirical results suggest that the proposed model outperforms competitors in terms of economic value.\n\\section*{Acknowledgments}\n\\noindent This research is supported by the Italian Ministry of Research PRIN 2013--2015, ``Multivariate Statistical Methods for Risk Assessment'' (MISURA), and by the ``Carlo Giannini Research Fellowship'', the ``Centro Interuniversitario di Econometria'' (CIdE) and ``UniCredit Foundation''. We would like to express our sincere thanks to all the participants to the Bozen -- Risk School, Bozen, September 2015, for their constructive comments that greatly contributed to improving the final version of the paper. A special thank goes also to Juri Marcucci and to all the participants of the 2015 Workshop in Econometrics and Empirical Economics (WEEE) held at the SADiBA, Perugia 27--28 August 2015, organised by the CIdE and the Bank of Italy.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLarge number of inflationary cosmology models is based on scalar fields, which \nplay an important role in the particle physics theories. The inflation is \nproduced by an homogeneous scalar field, dubbed inflaton, which under suitable \nconditions may lead to an early-time accelerated expansion. Following the first \nproposal of Guth~\\cite{Guth} and Sato~\\cite{Sato}, in the last years many \ninflationary models based on scalar fields ( and inspired by modified gravity \ntheories, string theories, quantum effects in the hot universe, etc) have been \nproposed.\n\nTypically the magnitude of scalar field is very large at the beginning of the \ninflation and\nthen it rolls down towards a potential minimum where the inflation ends (see \nRef.~\\cite{chaotic} as an example of chaotic inflation). In other models the \nfield can fall in a potential hole where it starts to oscillate and the \nrehating processes take place~\\cite{buca1, buca2, buca3, buca4}.\nSome more complicated models are based on a phase transition between two scalar \nfields: they are the so-called hybrid or double inflation models~\\cite{ibrida1, \nibrida2}.\nFor the introduction to the dynamics of inflation see Ref.~\\cite{Linde} and \nRefs.~\\cite{revinflazione}--\\cite{rr2}.\n\nRecently, cosmological and astrophysical data~\\cite{cosmdata} seem to confirm \nthe predictions of Starobinsky inflationary model~\\cite{Staro}. Such a model is \nbased on the account of $R^2$-term as the correction in the Einstein equations. \nThis quadratic correction emerges in the Planck epoch and plays a fundamental \nrole in the high curvature limit, when the early-time acceleration takes place. \nSuch theory is conformally equivalent to a scalar-tensor theory in the Einstein \nframe, where the inflaton drives the expansion in a quasi-de Sitter space-time \nand slowly moves to the end of inflation, when the reheating \nprocesses~\\cite{r1,r2,r3} start.\nSuch inflationary model has been recently revisited in many works. Among them, \nin Ref.~\\cite{genStar} a superconformal generalization of such a model in \nsuperconformal theory has been investigated, and in \nRefs.~\\cite{SS1}--\\cite{SS2} other applications based on the spontaneous \nbreaking of conformal invariance\nand on the scale-invariant extensions of Starobinsky model have been presented.\nIn Ref.~\\cite{cinesi}, a generalization of the Starobinsky model represented by \na polynomial correction of the Einstein gravity of the type $c_1 R^2+c_2 R^n$ \nhas been studied.\n\nIn this paper, we will concentrate on inflation caused by scalar curvature \ncorrections to Einstein gravity, namely,\nwe will consider the so-called $F(R)$-gravity, whose action is in the form of \n$F(R)=R+f(R)$, being $f(R)$ a function of the Ricci scalar (for recent reviews \non modified gravity,\nsee Refs.~\\cite{Review-Nojiri-Odintsov} --\\cite{SebRev} and \nRef.\\cite{others}). This kind of corrections may occur due to quantum effects \nin the\nhot universe or maybe motivated by the ultraviolet completation of quantum \ntheory of gravity. Our aim is to investigate which kinds of viable inflation \ncan be realized in the contest of $F(R)$-gravity beyond the Starobinsky model, \nwhose dynamics is governed in the Einstein frame by a potential of the type \n$V(\\sigma)\\sim (1-\\exp[-\\sigma])^2$, being $\\sigma$ the inflaton.\n\nThe paper is organized in the following way. In Sections {\\bf 2}--{\\bf 3}, we \nwill revisit the conformal transformations which permit to pass from the Jordan \nframe to the Einstein one and we will recall the dynamics of the viable \ninflation. In Section {\\bf 4}, we will study inflation for the general class of \nscalar potentials of the type $V(\\sigma)\\sim\\exp[n\\sigma]$, $n$ being a general \nparameter,\nperforming the analysis in the Einstein frame and therefore reconstructing the \n$F(R)$-gravity theories which correspond to the given potentials. Viable \ninflation must be consistent with the last Planck data (spectral index, \ntensor-to-scalar ratio...) and must correspond to Einstein theory corrections \nwhich emerge only at mass scales larger than the Planck one, namely at high \ncurvatures. We will see the conditions on $n$ for which slow-roll conditions \nare satisfied, and we will reconstruct the form of the $F(R)$-models during \nearly-time acceleration and their form at small curvaures. Some specific \nexamples are presented.\nIn Section {\\bf 5}, following the recent success of\nhigher derivative gravity, we will revisit and study in detail the specific \nclass of models $F(R)=R+(R+R_0)^n$. The analysis in the\nEinstein frame reveals that $n$ must be very close to two in order to realize a \nviable inflation for large and negative values of the scalar field,\nbut other possibilities are allowed\nby bounding the field in a different way.\nIn particular, when $n>2$, the de Sitter solution emerges, but in order to \nstudy the exit from inflation is necessary to analyze the theory in the Jordan \nframe, where perturbations make possible an early time acceleration with a \nsufficient amount of inflation.\nSome summary and outlook are given in Section {\\bf 6}.\nTechnical details and further considerations are presented in Appendixes.\n\n\n\\paragraph*{} We shall use units in which $c=\\hbar=k_{\\mathrm{B}}=1$,\n$c\\,,\\hbar\\,,k_{\\mathrm{B}}$ being respectively the speed of light, the\nPlanck and Boltzmann constants. Moreover we shall denote by $G_N$\nthe gravitational constant and by $M_{\\mathrm{Pl}} =G_{N}^{-1\/2} =1.2\n\\times 10^{19}$GeV the Planck mass.\nFinally, we shall set $\\kappa^2\\equiv 8 \\pi G_{N}$.\n\n\n\n\\section{Conformal transformations}\n\nIn scalar-tensor theories of gravity, a scalar field coupled to the metric \nappears in the action.\nThe first scalar-tensor theory was proposed by Brans \\& Dicke in \n1961~\\cite{BransDicke} in the attempt to incorporate the Mach's principle into \nthe theory of gravity, but today the interest to such theories is related with \nthe possibility to reproduce the primordial acceleration of the inflationary \nuniverse.\n\nIn principle, a modified gravity theory can be rewritten in scalar-tensor or \nEinstein frame\nform.\nLet us start by considering the general action of $F(R)$-modified gravity\n\\begin{equation}\nI = \\int_\\mathcal{M} d^4 x \\sqrt{-g} \\left[ \\frac{F(R)}{2\\kappa^2}\\right]\\,,\n\\label{action}\n\\end{equation}\nwhere $F(R)$ is a function of the Ricci scalar $R$, $g$ is the determinant of \nthe metric tensor $g_{\\mu\\nu}$ and\n$\\mathcal{M}$ is the space-time manifold.\nNow we introduce the field $A$ into\n~(\\ref{action}),\\\\\n\\phantom{line}\n\\begin{equation}\nI_{JF}=\\frac{1}{2\\kappa^{2}}\\int_{\\mathcal{M}}\\sqrt{-g}\\left[\nF_A(A) \\, (R-A)+F(A)\\right] d^{4}x\\label{JordanFrame}\\,.\n\\end{equation}\n\\phantom{line}\\\\\nHere, `$JF$' means `Jordan frame', and $F_A(A)$ denotes the derivative of \n$F(A)$ with respect to $A$. By making the variation\nwith respect to $A$, we immediatly obtain $A=R$, such that (\\ref{JordanFrame}) \nis equivalent to (\\ref{action}) .\nWe define the scalar field $\\sigma$ (which in fact encodes the new degree of \nfreedom in the theory, namely the scalaron or inflaton) as\n\\begin{equation}\n\\sigma := -\\sqrt{\\frac{3}{2\\kappa^2}}\\ln [F_A(A)]\\,.\\label{sigma}\n\\end{equation}\nBy considering the conformal transformation of the metric,\n\\begin{equation}\n\\tilde g_{\\mu\\nu}=\\mathrm{e}^{-\\sigma}g_{\\mu\\nu}\\label{conforme}\\,,\n\\end{equation}\nwe finally get the `Einstein frame' action\n\\phantom{line}\n\\begin{eqnarray}\nI_{EF} &=& \\int_{\\mathcal{M}} d^4 x \\sqrt{-\\tilde{g}} \\left\\{ \n\\frac{\\tilde{R}}{2\\kappa^2} -\n\\frac{1}{2}\\left(\\frac{F_{AA}(A)}{F_A(A)}\\right)^2\n\\tilde{g}^{\\mu\\nu}\\partial_\\mu A \\partial_\\nu A - \n\\frac{1}{2\\kappa^2}\\left[\\frac{A}{F_A(A)}\n- \\frac{F(A)}{F_A(A)^2}\\right]\\right\\} \\nonumber\\\\ \\nonumber\\\\\n&=&\\int_{\\mathcal{M}} d^4 x \\sqrt{-\\tilde{g}} \\left( \n\\frac{\\tilde{R}}{2\\kappa^2} -\n\\frac{1}{2}\\tilde{g}^{\\mu\\nu}\n\\partial_\\mu \\sigma \\partial_\\nu \\sigma - \nV(\\sigma)\\right)\\,,\\label{EinsteinFrame}\n\\end{eqnarray}\n\\phantom{line}\\\\\nwhere $\\tilde{R}$ denotes the Ricci scalar evaluated in the conformal metric \n$\\tilde{g}_{\\mu\\nu}$ and $\\tilde{g}$ is the determinant of the conformal \nmetric, namely $\\tilde{g}=\\mathrm{e}^{-4\\sigma}g$.\nFurthermore, one has\\\\\n\\phantom{line}\n\\begin{equation}\nV(\\sigma)\\equiv\\frac{A}{F'(A)} - \n\\frac{F(A)}{F'(A)^2}=\\frac{1}{2\\kappa^2}\\left\\{\\mathrm{e}^{\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\nR\\left(\\mathrm{e}^{-\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\\right)\n-\\mathrm{e}^{2\\left(\\sqrt{2\\kappa^2\/3}\\right) \n\\sigma}F\\left[R\\left(\\mathrm{e}^{-\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\\right)\\right]\n\\right\\}\\,,\\label{V}\n\\end{equation}\n\\phantom{line}\\\\\nbeing $R(\\mathrm{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma})$ the solution of\nEq.~(\\ref{sigma}) with $A=R$, namely $R$ a function of \n$\\mathrm{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}$.\nIn what follows, we will omit the tilde to denote all the quantities evaluated \nin the Einstein frame.\n\nNote that string-inspired inflationary models also contain canonical \nand\/or tachyon scalar (for recent discussion see Refs.~\\cite{Samiinfl}--\\cite{Jamil:2013nca}). \n\n\n\\section{Dynamics of inflation}\n\nIn this Section, for the sake of completeness, we will recall the well-known \nfacts on inflation. The energy density and pressure of the inflaton $\\sigma$ \nare given by\n\\begin{equation}\n\\rho_\\sigma=\\frac{\\dot\\sigma^2}{2}+V(\\sigma)\\,,\\quad \np_\\sigma=\\frac{\\dot\\sigma^2}{2}-V(\\sigma)\\,,\n\\end{equation}\nwhere the dot is the derivative with respect to the cosmological time. The \nFriedmann equations in the presence of $\\sigma$ read\n\\begin{equation}\n\\frac{3 \nH^2}{\\kappa^2}=\\frac{\\dot\\sigma^2}{2}+V(\\sigma)\\,,\\quad-\\frac{1}{\\kappa^2}\\left(2\\dot \nH+3 H^2\\right)=\\frac{\\dot\\sigma^2}{2}-V(\\sigma)\\,,\n\\end{equation}\nand the energy conservation law coincides with the equation of motion for \n$\\sigma$ and reads\n\\begin{equation}\n\\ddot\\sigma+3H\\dot\\sigma=-V'(\\sigma)\\,,\n\\end{equation}\nwhere the prime denotes the derivative of the potential with respect to \n$\\sigma$.\nFrom the Friedmann equations we obtain\n\\begin{equation}\n\\dot{H}=-\\frac{\\kappa^2}{2}\\left( \\rho_\\sigma+ p_\\sigma \n\\right)=-\\frac{\\kappa^2}{2} \\dot\\sigma^2\\,.\n\\end{equation}\nOn the other hand, the acceleration can be expressed as\n\\begin{equation}\n\\frac{\\ddot{a}}{a}= H^2+\\dot{H}=H^2\\left(1-\\epsilon \\right)\\,,\n\\end{equation}\nwhere we have introduced the ``slow roll'' parameter\n\\begin{equation}\n\\epsilon=-\\frac{\\dot{H}}{H^2}\\,.\n\\end{equation}\nThis parameter may be expressed as a function of the inflaton as\n\\begin{equation}\n\\epsilon=\\frac{\\kappa^2 \\dot\\sigma^2}{2 H^2} \\,.\n\\end{equation}\nWe also have\n\\begin{equation}\n\\frac{\\ddot{a}}{a}=\\frac{\\kappa^2}{3}\\left(V-\\dot\\sigma^2 \\right)\\,.\n\\end{equation}\nThus, the condition to have an acceleration is $\\epsilon <1$ or $\\dot\\sigma^2 \n< V(\\sigma)$. There is another slow roll parameter defined by\n\\begin{equation}\n\\eta=-\\frac{\\ddot{H}}{2H\\dot{H}}=\\epsilon-\\frac{1}{2 \\epsilon H}\\dot\\epsilon \n\\,.\n\\end{equation}\nAs a function of the inflaton one has\n\\begin{equation}\n\\eta=-\\frac{\\ddot\\sigma}{H\\dot\\sigma} \\,.\n\\end{equation}\nFor the inflation to occur and persist for a convenient amount of time, a quasi \nde Sitter space is required, namely $\\dot H$ has to be very small, and, as a \nresult, also the two slow roll parameters have to be very small, and one has\n\\begin{equation}\n\\dot\\sigma^2\\ll V(\\sigma)\\,,\n\\end{equation}\nnamely the kinetic energy of the field has to be small during the inflation. As \na result, the Friedmann equations reduce to\n\\begin{equation}\n\\frac{3H^2}{\\kappa^2}\\simeq V(\\sigma)\\,,\n\\quad\n3H\\dot\\sigma\\simeq -V'(\\sigma)\\,.\\label{EOMs}\n\\end{equation}\nIt is easy to show that within this slow roll regime,\n the slow roll parameters may be expressed as function of the inflaton \npotential as\n\\begin{equation}\n\\epsilon=\\frac{1}{2\\kappa^2}\\left(\\frac{V'(\\sigma)}{V(\\sigma)}\\right)^2\\,,\\quad \n\\eta=\\frac{1}{\\kappa^2}\\left(\\frac{V''(\\sigma)}{V(\\sigma)}\\right)\\,.\\label{slow}\n\\end{equation}\nInflation ends when $\\epsilon\\,,|\\eta|\\sim 1$. A useful quantity which \ndescribes the amount of inflation is e-foldings number $N$ defined by\n\\begin{equation}\nN\\equiv\\ln \\frac{a_f}{a_i}=\\int_{t_i}^{t_e} H dt\\simeq \n\\kappa^2\\int^{\\sigma_i}_{\\sigma_e}\\frac{V(\\sigma)}{V'(\\sigma)}d\\sigma\\,,\\label{N}\n\\end{equation}\nwhere the indices $i,f$ are referred to the quantities at the beginning and the \nend of inflation, respectively. The required e-foldings number for inflation is \nat least $N\\simeq 60$.\nThe amplitude of the primordial scalar power spectrum is\n\\begin{equation}\n\\Delta_{\\mathcal R}^2=\\frac{\\kappa^4 V}{24\\pi^2\\epsilon}\\,,\\label{spectrum}\n\\end{equation}\nand for slow roll inflation the spectral index $n_s$ and the tensor-to-scalar \nratio are given by\n\\begin{equation}\nn_s=1-6\\epsilon+2\\eta\\,,\\quad r=16\\epsilon\\,.\\label{indexes}\n\\end{equation}\nThe last Planck data constrain these quantities as\n\\begin{equation}\nn_s=0.9603\\pm0.0073\\,,\\quad r<0.11\\,.\\label{data}\n\\end{equation}\n\n\n\\section{Reconstruction of $F(R)$ theory from the scalar potential and analysis \nof the inflation in the Einstein frame}\n\nIn this Section, we\nwill study some classes of scalar potential which produce inflation in the \nEinstein frame. The aim is to generalize the Starobinsky model\nby considering different behaviour of the scalar potential as \n$V(\\sigma)\\sim\\exp[n\\sqrt{2\\kappa^2\/3}\\sigma]$, where $n$ is the parameter on \nwhich it depends the dynamics of the inflation (slow roll parameters, spectral \nindices...). This analysis is motivated by the possibility to reconstruct \nsuitable $F(R)$ corrections to General Relativity in the corresponding Jordan \nframe. By `suitable' we mean corrections that vanish at mass scales smaller \nthan the Planck mass $M_{\\text{Pl}}$ and give rise to corrections only in the high \ncurvature limits, namely during the inflationary period.\nEvery scalar potential will be confronted with cosmological data.\n\nIn order to reconstruct the $F(R)$-gravity which corresponds to a given \npotential, we may start from Eq.~(\\ref{V}). By dividing such equation to \n$\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\right]$, and then by taking the derivative with respect to \n$R$, we get\n\\begin{equation}\nR \nF_R(R)=-2\\kappa^2\\sqrt{\\frac{3}{2\\kappa^2}}\\frac{d}{d\\sigma}\\left(\\frac{V(\\sigma)}{e^{2\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}}\\right)\\,.\\label{start}\n\\end{equation}\nAs a result, giving the explicit form of the potential $V(\\sigma),$ thanks to \nthe relation (\\ref{sigma}), we obtain an equation for $F_R(R)$, and therefore \nthe $F(R)$-gravity model in the Jordan frame. In this process, one introduces \nthe integration constant, which has to be fixed by\nrequiring that Eq.~(\\ref{V}) holds true.\n\n\\subsection{$V(\\sigma)\\sim c_0+c_1\\exp[\\sigma]+c_2\\exp[2\\sigma]$: $R+ \nR^2+\\Lambda $-models\\label{4.1}}\n\nLet us start with the following inflationary potential\n\\begin{equation}\nV(\\sigma)=\\left[c_0+c_1\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}+c_2\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}\n\\right]\\,.\\label{cucu}\n\\end{equation}\nThis is the simplest example and is the minimal generalization of the \nStarobinsky model.\nEquation (\\ref{start}) gives\n\\begin{equation}\n2c_0F_R^2+c_1F_R-RF_R=0\\,.\\label{req}\n\\end{equation}\nAssuming $F_R \\neq 0$, one gets\n\\begin{equation}\nF_R(R)=-\\frac{c_1}{2c_0}+\\frac{R}{2c_0}\\,.\\label{FF}\n\\end{equation}\nThus, the corresponding Lagrangian of $F(R)$ gravity is given by\n\\begin{equation}\nF(R)=-\\frac{c_1}{2c_0}R+\\frac{R^2}{4c_0}+\\Lambda\\,.\\label{Staro2}\n\\end{equation}\nHere, $\\Lambda$ is a constant of integration, which can be determined by \nequation (\\ref{V}). The result is\n\\begin{equation}\n\\Lambda=\\frac{c_1^2}{4c_0}-c_2\\,.\\label{L}\n\\end{equation}\nAn important remark is in order. In order to have the correct Einstein-Hilbert \nterm, we must put $-c_1\/(2c_0)=1$. Thus, we have the class \nof modified quadratic models depending on two constants\n\\begin{equation}\nF(R)=R+\\frac{R^2}{4c_0}+c_0-c_2\\,.\\label{Staro3}\n\\end{equation}\nFurthermore, in the specific case $c_2=0$, one has\n \\begin{equation}\nF(R)=R+\\frac{R^2}{4c_0}+ c_0\\,.\\label{cogno}\n\\end{equation}\nThis is an interesting model, and in the Appendix B, we will study its static \nsperically symmetric solutions.\n\nThe other interesting case is the vanishing of cosmological constant, namely\n\\begin{equation}\nc_2=c_0.\\label{L1}\n\\end{equation}\nSince we are assuming $c_1\/(2c_0)=1$, it also follows \n$c_2=-c_1\/2$. As a consequence, we recover the\nStarobinsky model $F(R)=R+\\frac{R^2}{4c_0}$, and related Einstein frame \npotential\n\\begin{equation}\nV(\\sigma)=c_0 \\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\n\\right)^2\\,.\\label{VStaro}\n\\end{equation}\nThe model (\\ref{Staro3}) is the extension of the Starobinsky model to the case \nwith cosmological constant and gives a viable inflation.\nIn what follows, we denote\n\\begin{equation}\nc_0=\\frac{\\gamma}{4\\kappa^2}\\,.\n\\end{equation}\nThe initial value of the inflaton is large (and negative) and it rolls down \ntoward the potential minimum at $\\sigma\\rightarrow 0^-$, \n$V(0)=-\\gamma\/(4\\kappa^2)<0$. During inflation ($\\sigma\\rightarrow-\\infty$) \nequations (\\ref{EOMs}) lead to\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{12}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{\\gamma}{\\sqrt{6\\kappa^2}}\\right)\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\\label{copy}\n\\end{equation}\nIt means, that a quasi de Sitter solution can be realized.\nWe must require\n\\begin{equation}\n\\gamma\\propto M^2\\,,\\quad M\\ll M_P\\,,\\label{gammacond}\n\\end{equation}\nwhere $M_P$ is the Planck mass. As a consequence, the corrections of Eintein\ngravity emerge at high curvature and one gets the accelerated expansion with \n$H\\propto\\sqrt{\\gamma}$.\nThe scalar field behaves as\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{3}{2\\kappa^2}}\\ln\\left[\\frac{1}{3}\\sqrt{\\frac{2\\gamma}{3}}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. If at this time \n$|\\sigma|$ is very large, the slow roll paramters (\\ref{slow}),\n\\begin{equation}\n\\epsilon=\\frac{4}{3}\\frac{1}{(2-\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma})^2}\\simeq \n0\\,,\\quad|\\eta|=\\frac{4}{3}\\frac{1}{|2-\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}|}\\simeq \n0\\,,\\label{star}\n\\end{equation}\nare very small and the field moves slowly.\nThe inflation ends when such parameters are of the order of unit, namely at\n$\\sigma_e\\simeq-0.17\\sqrt{3\/(2\\kappa^2)}$.\nThe e-foldings number can be evaluated from (\\ref{N}) and reads \n($\\sigma_i\\gg\\sigma_e$),\n\\begin{equation}\nN\\simeq\\frac{3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}}{4}\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\\frac{1}{4}\\sqrt{\\frac{2\\gamma}{3}}t_0\\,.\n\\end{equation}\nThe inflation ends at $t_e=t_0-3\\sqrt{3\/(2\\alpha)}\\exp[0.17]$. For example, in \norder to obtain $N=60$, we must require \n$\\sigma_i\\simeq-\\sqrt{3\/(2\\kappa^2)}4.38\\simeq 1.07 M_{pl}$. Moreover, we may \nexpress the e-foldings numbers as\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{4N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nThe amplitude of primordial power spectrum (\\ref{spectrum}) is\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma \nN^2}{72\\pi^2}\\propto\\frac{\\kappa^2 M^2 N^2}{72\\pi^2}\\ll \\frac{\\kappa^2M_P^2 \nN^2}{72\\pi^2}\\,,\n\\end{equation}\nand the indexes (\\ref{indexes}) result to be\n\\begin{equation}\nn_s\\simeq1-\\frac{2}{N}\\,,\\quad r\\simeq\\frac{12}{N^2}\\,.\n\\end{equation}\nSince we have $n_s>1-\\sqrt{0.11\/3}\\simeq 0.809$ when $r<0.11\\,,n_s<1$, we see \nthat these indices are compatible with (\\ref{data}). For example, for $N=60$, \none has $n_s=0.967$ and $r=0.003$.\nWe stress that this behaviour is the one of Starobinsky model, with a different \nminimum of the potential and a cosmological constant in the Jordan frame. The \nappeareance of a cosmological constant at large curvature needs some \nexplanation. It can be originated by some quantum effects or may be supported \nby a modified gravity term which makes it to vanish at small curvatures (see \nfor examples the so called `two step-models' in \nRefs.~\\cite{twostep1}--\\cite{twostep2}). However, if the cosmological constant \nis set equal to zero (or, if necessary, is set equal to an other value), the feature of the model in Einstein frame does not \nchange during the inflation: we will see that this double possibility, namely, \ntaking a cosmological constant in the Jordan frame, or taking an additional \nterm $\\propto\\exp[2\\sqrt{2\\kappa^2\/3}\\sigma]$ in the Einstein frame, does not \nmodify the proprieties of the scalar potentials during the early time \nacceleration.\n\n\n\\subsection{$V(\\sigma)\\sim\\gamma\\exp[-n\\sigma]\\,,n>0$: $c_0 \nR^{\\frac{n+2}{n+1}}$-models\\label{case}}\n\nAs a second example, we consider the following potential\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}\\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\right)+\\frac{\\gamma}{\\kappa^2}\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\n\\end{equation}\nbeing $\\gamma\\,,n>0$ constants. This potential possesses a minimum in which the \nscalar field may fall at the end of inflation. Since for large and negative \nvalues of the scalar field the potential is not flat, we do not expect a de \nSitter universe, but if the slow roll conditions are\nsatisfied, we can obtain an acceleration with a sufficient amount of inflation.\n\nBy using our reconstruction,\nfrom (\\ref{start}) one derives\n\\begin{equation}\nF_R(R)-\\frac{1}{2}+\\frac{\\gamma}{2\\alpha}(2+n) \nF_R(R)^{1+n}=\\frac{R}{4\\alpha}\\,.\\label{pippo}\n\\end{equation}\nIn principle, this equation admitts many solutions. At the perturbative level, \nit is easy to find that, by putting\n\\begin{equation}\n\\alpha=-\\gamma(n+2)\\,,\\label{alphacond}\n\\end{equation}\nwe obtain\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq 1+c_1 R+c_2 R^2+c_3 R^3+...\\,,\\label{funzioncina}\n\\end{equation}\nwhen $|c_1 R|\\,,|c_2 R^2|\\,, |c_3 R^3|...\\ll 1$ (see Appendix A). It means \nthat, since \n$c_1\\propto\\gamma^{-1}\\,,c_2\\propto\\gamma^{-2}\\,,c_3\\propto\\gamma^{-3}...$,\nif $\\gamma$ satisfies (\\ref{gammacond}), we recover the Einstein's gravity when \n$R\\ll\\gamma$ and the theory is an high curvature correction to General \nRelativity. Moreover, when $R\\gg\\gamma$, the asymptotic solution of \nEq.~(\\ref{pippo}) with (\\ref{alphacond}) is given by\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq\\left(\\frac{1}{4(n+2)}\\right)^\\frac{1}{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{1}{n+1}}\\,,\\quad\nF(R\\gg\\gamma)\\simeq\\gamma\\left(\\frac{n+1}{n+2}\\right)\\left(\\frac{1}{4(n+2)}\\right)^\\frac{1}{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{n+2}{n+1}}\\,.\\label{expression}\n\\end{equation}\nHere, one important comment is required. In the Einstein frame, \n$R_{EF}\\sim\\gamma$ during inflation, but the corresponding curvature in the \nJordan frame is $R_{JF}\\simeq \\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}R_{EF}$ (we \nmay neglect the kinetic energy of scalar field in the slow roll approximation), \nsuch that $R_{JF}\\gg\\gamma$ when $\\sigma\\rightarrow-\\infty$ and expression \n(\\ref{expression}), which is evaluated in the Jordan frame, effectively is \nvalid for inflation\n(for $n\\rightarrow 0$ we recover $F(R)\\sim R^2$ in the Jordan frame).\n\nThe potential finally reads\n\\begin{equation}\nV(\\sigma)=-\\frac{\\gamma(n+2)}{\\kappa^2}\\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\right)+\\frac{\\gamma}{\\kappa^2}\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\n\\end{equation}\nThis potential has a minimum ($V'(\\sigma_{min})=0$) at \n$\\sigma_{min}=-\\sqrt{3\/(2\\kappa^2)}\\log[(n+2)\/n]\/(n+1)$, and one gets\n\\begin{equation}\nV(\\sigma_{min})=\\frac{\\gamma}{\\kappa^2}\\left(n^{\\frac{1}{n+1}}(n+2)^{\\frac{n}{n+1}}+\\left(\\frac{n+2}{n}\\right)^{\\frac{1}{n+1}}-(n+2)\\right)>0\\,,\\quad\\gamma\\,,n>0\\,.\n\\end{equation}\nWhen $\\sigma\\rightarrow-\\infty$ (large curvature), the potential goes to \ninfinity, and when $\\sigma\\rightarrow 0^-$, $V(\\sigma)=\\gamma\/\\kappa^2$.\nSince the slow roll parameters are given by\n\\begin{eqnarray}\n\\epsilon&=&\\frac{\\left(n-(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}\\right)^2}{3\\left(\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}-(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}-1\n\\right)^2}\\,,\\nonumber\\\\ \\nonumber\\\\\n|\\eta|&=&\\frac{2}{3}\\frac{n^2+(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}}{|1+(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}-\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\n|}\\,,\n\\end{eqnarray}\none has $\\epsilon(\\sigma\\rightarrow-\\infty)\\simeq n^2\/3$ and\n$|\\eta(\\sigma\\rightarrow-\\infty)|\\simeq2n^2\/3$, which implies $00\\,,\\quad (n<\\sqrt{3})\n\\end{equation}\nand we have an acceleration as soon as $\\epsilon<1$. Despite to the fact that \nin this kind of models the acceleration is smaller than in the de Sitter \nuniverse, the slow roll paramters can be small enough\nto justify our slow roll approximations. A direct evaluation of the ratio of \nkinetic energy of the field and potential leads to \n$\\left(\\dot\\sigma^2\/2\\right)\/V(\\sigma)=n^2\/9$, which is much smaller than one \nwhen $n\\ll 1$. The inflation ends when the slow roll paramters are on the order \nof unit, before the minimum of the potential. Note that, by definition, since \n$V'(\\sigma_{min})=0$, one has $\\epsilon(\\sigma_{min})=0$, which corresponds to \nthe minimum of the slow roll paramter. However, before to this point, since the \nslow roll parameters behave as\n\\begin{equation}\n\\epsilon\\simeq\\frac{n^2}{3\\left(\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}-1\n\\right)^2}\\,,\\quad\n|\\eta|\\simeq\\frac{2}{3}\\frac{n^2}{|1-\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\n|}\\,,\n\\end{equation}\nwe find that $\\epsilon\\,,|\\eta|\\simeq 1$ when \n$\\sigma\\simeq-\\sqrt{3\/(2\\kappa^2)}\\log[(6+3n)\/(3-n\\sqrt{3})]\/n$.\nFinally, from Eq.~(\\ref{N}), we get the $N$-foldings number of inflation,\n\\begin{equation}\nN\\simeq -\\frac{1}{n}\\sqrt{\\frac{3\\kappa^2}{2}}\\sigma\n\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq-\\frac{3}{n^2}\\ln\\left[\\frac{n^2}{3}\\sqrt{\\frac{\\gamma}{3}}t_0\\right]\\,.\n\\end{equation}\nWhen the inflation ends, the field falls in the minimum of the potential and \nstarts to oscillate.\nThe reheating process takes place. The amplitude of primordial power spectrum \n(\\ref{spectrum}) and the spectral indexes (\\ref{indexes}) can be written as\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma \n\\text{e}^{\\frac{2}{3}N\\,n^2}}{8\\pi^2n^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{2n^2}{3}\\,,\\quad r\\simeq\\frac{16n^2}{3}\\,.\n\\end{equation}\nThe corrections to $n_s$ and $r$ are on the order of $\\exp\\left[-2n^2 \nN\/3\\right]\\ll 1$.\nThese indexes are compatible with (\\ref{data}) when\n\\begin{equation}\nn\\sim\\frac{1}{10}\\,,\\frac{2}{10}\\,.\n\\end{equation}\nThese are the typical values of $n$ which make the scalar potential (\\ref{V3}) \nable to reproduce a viable inflationary scenario. The result suggests that only \nthe models\nclose to $R^2$-gravity are able to produce this kind of inflation.\n\nIn general, $F_R(R)$ in Eq.~(\\ref{funzioncina}) may lead to a cosmological \nconstant proportional to $\\gamma$ in the $F(R)$-model. However, we can set it \nequal to zero, adding a suitable term in\nthe potential ($\\ref{V3}$) proportional to\n$\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, which\nchanges much slowler than\n$\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ when $00$, as usually, constants. It follows\n\\footnote{Here, we exclude the solution with the minus sign in front of the \nsquare root which leads to an imaginary value of $\\sqrt{F_R}$ when $R$ is \nreal.}\n\\begin{equation}\nF_R(R)=\\frac{9\\gamma^2+8R\\alpha+3\\sqrt{16 \nR\\alpha\\gamma^2+9\\gamma^4}}{32\\alpha^2}\\,.\n\\end{equation}\nSince we must require $\\alpha\\,,\\gamma\\gg 1$ and at small curvature we want to \nrecover the Einstein gravity ($F_R=1$), we set\n$\\alpha=3\\gamma\/4$, $\\gamma>0$, and we get\n\\begin{equation}\nF_R(R)=\\frac{1}{2}+\\frac{1}{3\\gamma}R+\\frac{\\sqrt{3}}{6}\\sqrt{4R\/\\gamma+3}\\,.\\label{FFFF}\n\\end{equation}\nThus, from Eq.~(\\ref{V}) we obtain\n\\begin{equation}\nF(R)=\\frac{R}{2}+\\frac{R^2}{6\\gamma}+\\frac{\\sqrt{3}}{36}\\left(4R\/\\gamma+3\\right)^{3\/2}+\\frac{\\gamma}{4}\\,.\\label{zippozap}\n\\end{equation}\nHere, we stress that the conformal transformation gives for the Ricci scalar\n\\begin{equation}\nR=3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}\\left(1+\n\\text{e}^{\\sqrt{2\\kappa^2\/3\/2}\\sigma\/2}\\right)\\,,\\,\n3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}\\left(1-\n\\text{e}^{\\sqrt{2\\kappa^2\/3\/2}\\sigma\/2}\\right)\\,,\n\\end{equation}\nbut only the second one leads to our potential (namely, is the one wich emerges \nfrom our reconstruction). For $R\\ll\\gamma$, the model reads $F(R\\ll\\gamma)\\simeq R+\\gamma\/2$. If we want to recover the General Relativity action $F(R\\ll\\gamma)\\simeq R$, we must set the cosmological constant, namely the last term in (\\ref{zippozap}), equal to $-\\gamma\/4$: in this case\nthe scalar potential is\n\\begin{equation}\nV(\\sigma)=\\frac{3\\gamma}{4\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}\n-\\gamma\\frac{\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}}{4\\kappa^2}\\,.\\label{V2bis}\n\\end{equation}\nLet us analyze the possibility to reproduce inflation from the potential \n(\\ref{V2}), which finally reads\n\\begin{equation}\nV(\\sigma)=\\left(\\frac{3}{4\\kappa^2}-\\frac{\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}}\n{\\kappa^2}\\right)\\gamma\\,.\n\\end{equation}\nThe initial value of the inflaton is large (and negative) and it rolls down \ntoward the potential minimum at $\\sigma\\rightarrow 0^-$, \n$V(0)=-\\gamma\/(4\\kappa^2)<0$. When $\\sigma\\rightarrow-\\infty$ the EOMs in the \nslow roll limit read\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{4}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{\\gamma}{\\sqrt{6\\kappa^2}}\\right)\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}\\,.\n\\end{equation}\nIt means, that $\\gamma$ must satisfy condition (\\ref{gammacond}), such that the \ninflation takes place at the Plank epoch.\nThe de Sitter expansion can be realized and the field behaves as\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{6}{\\kappa^2}}\\ln\\left[\\frac{\\sqrt{\\gamma}}{9}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. If at this time the \nmagnitude of $\\sigma$ is very large, the slow roll parameters (\\ref{slow})\n\\begin{equation}\n\\epsilon=\\frac{4}{3}\\frac{1}{\\left(4-3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}\\right)^2}\\simeq\n0\\,,\\quad|\\eta|=\\frac{2}{3}\\frac{1}{|4-3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}|}\\simeq \n0\\,,\n\\end{equation}\nare small and the field moves slowly.\nThe inflation ends at\n$\\sigma_e\\simeq-0.12\\sqrt{3\/(2\\kappa^2)}$, when the slow roll parameters are of \nthe order of unit.\nThe e-foldings number can be evaluated from (\\ref{N}) and read,\n\\begin{equation}\nN\\simeq\\frac{9\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}}{2}\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\\frac{\\sqrt{\\gamma}}{2}t_0\\,.\n\\end{equation}\nAs a consequence, the slow roll parameters can be written as\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nThe amplitude of primordial power spectrum (\\ref{spectrum}) and the spectral \nindexes (\\ref{indexes}) are given by\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma N^2}{96\\pi^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{1}{N}\\,,\\quad r\\simeq\\frac{48}{N^2}\\,.\n\\end{equation}\nSince from these formulas we have $n_s>1-\\sqrt{0.11\/48}\\simeq 0.9521$ when \n$r<0.11\\,,n_s<1$, we see that these expressions are compatible with \n(\\ref{data}). For example, for $N=60$, one has $n_s=0.967$ and $r=0.013$.\n\nWe finish this Subsection with some considerations on the potential \n($\\ref{V2bis}$), which corresponds to the model with cosmological constant equal to $-\\gamma\/4$. \nSince the term $\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ changes slower than\n$\\exp\\left[\\sqrt{2\\kappa^2\/3}\\sigma\/2\\right]$, the dynamics of inflation is the \nsame of above. In this case, when the inflaton exits from the slow roll region, \nit falls in the minimum of the potential located at $V(0)=-\\gamma\/(8\\kappa^2)$.\n\n\n\\subsection{$V(\\sigma)\\sim\\gamma(2-n)\/2-\\gamma\\exp[n\\sigma]\\,,00\\,,\\label{alphaalpha}\n\\end{equation}\nif $\\gamma$ satisfies (\\ref{gammacond}),\nat small curvature one gets\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq 1+c_1 R+c_2 R^2+c_3 R^3+...\\,,\\label{espansione}\n\\end{equation}\nwith \n$c_1\\propto\\gamma^{-1}\\,,c_2\\propto\\gamma^{-2}\\,,c_3\\propto\\gamma^{-3}...$, \nsuch that our theory is the high curvature correction to General Relativity \n(see Appendix A).\nFor example, in the previous Subsection we have seen an exact solution for the \ncase $n=1\/2$. Since $1\\ll \\gamma$, when $R\/\\gamma\\ll 1$ we can expand $F_R(R)$ \nin (\\ref{FFFF}) as\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq \n1+\\frac{2}{3\\gamma}R-\\frac{R^2}{9\\gamma^2}+...\\,,\\label{primeF}\n\\end{equation}\nwhich returns to be (\\ref{espansione}) by using the coefficients in Appendix A.\nOn the other side, when $R\\gg\\gamma$, the asymptotic solution of \nEq.~(\\ref{pippo2}) with (\\ref{alphaalpha}) is given by\n\\begin{eqnarray}\nF_R(R\\gg\\gamma)&\\simeq&\\left(\\frac{R}{2\\gamma(2-n)}\\right)+\\left(\\frac{R}{2\\gamma(2-n)}\\right)^{1-n}\\,,\\nonumber\\\\\\nonumber\\\\\nF(R\\gg\\gamma)&\\simeq&\\frac{1}{2}\\left(\\frac{R^2}{2\\gamma(2-n)}\\right)+\\frac{1}{2-n}\\left(\\frac{1}{2\\gamma(2-n)}\\right)^{1-n}R^{2-n}\\,.\n\\label{megastana}\n\\end{eqnarray}\nAlso in this case, by taking the exact solution of the prevoius Section in the \nhigh curvature limit,\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq\\frac{R}{3\\gamma}+\\sqrt{\\frac{R}{3\\gamma}}\\,,\n\\end{equation}\nwe can verify the consistence of expression (\\ref{megastana}) for $n=1\/2$.\n\nAs an other example, let us consider the case $n=1\/3$. The reconstruction leads \nto\\\\\n\\phantom{line}\\\\\n\\begin{equation}\nF_R(R)=\\frac{1}{3}+\\frac{3}{10}\\left(\\frac{R}{\\gamma}\\right)\n+\\frac{2}{3\\times \n5^{1\/3}}\\left[9\\left(\\frac{R}{\\gamma}\\right)+5\\right]\\frac{1}{\\Delta^{1\/3}}+\\frac{1}{6\\times\n5^{2\/3}}\\Delta^{1\/3}\\,,\n\\end{equation}\n\\phantom{line}\\\\\nwhere\n\\phantom{line}\n\\begin{equation}\n\\Delta=200+243\\left(\\frac{R}{\\gamma}\\right)^2+540\\left(\\frac{R}{\\gamma}\\right)+27\\sqrt{81\\left(\\frac{R}{\\gamma}\\right)^4+40\\left(\\frac{R}{\\gamma}\\right)^3}\\,.\n\\end{equation}\n\\phantom{line}\\\\\nAt small curvature, it is easy to find\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq1+\\frac{9}{10}\\left(\\frac{R}{\\gamma}\\right)+...\\,,\n\\end{equation}\nand in the high curvature limit this model has the following structure,\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq \n\\frac{3}{10}\\left(\\frac{R}{\\gamma}\\right)+\\left(\\frac{3}{10}\\right)^{\\frac{2}{3}}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{2}{3}}\\,,\n\\end{equation}\nwhich corresponds to (\\ref{megastana}) with $n=1\/3$.\nFinally, the model (\\ref{FF}) is the limiting case of $n\\rightarrow 1$.\n\nLet us analyze this class of potentials. When the magnitude of the inflaton is \nlarge, the scalar potential (\\ref{V3}) with\n$\\alpha=\\gamma(2-n)\/2$,\n\\begin{equation}\nV(\\sigma)=\\frac{\\gamma(2-n)}{2\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\label{n<2}\n\\end{equation}\nbehaves as $V(\\sigma)\\simeq\\gamma(2-n)\/(2\\kappa^2)$, and the EOMs in the slow \nroll limit read\n\\begin{equation}\nH^2\\simeq\\left(\\frac{\\gamma(2-n)}{6}\\right)\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{n\\gamma\\sqrt{2}}{\\sqrt{3\\kappa^2}}\\right)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\n\\end{equation}\nAs a consequence, the field results to be\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{3}{2\\kappa^2}}\\frac{1}{n}\\ln\\left[\\frac{2\\sqrt{2}}{3\\sqrt{3}}\\frac{n^2\\sqrt{\\gamma}}{\\sqrt{2-n}}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. When \n$\\sigma\\rightarrow-\\infty$, the slow roll parameters became\n\\begin{equation}\n\\epsilon=\\frac{4n^2}{3}\\frac{1}{\\left(2+(n-2)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\right)^2}\\simeq\n0\\,,\n\\quad\n|\\eta|=\\frac{4n^2}{3}\\frac{1}{|2+(n-2)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}|}\\simeq \n0\\,,\n\\end{equation}\nand are very small.\nThe inflation ends at\n$\\sigma_e\\simeq-(1\/n)\\sqrt{3\/(2\\kappa^2)}\\log\\left[(\\gamma\/\\alpha)(1-n\/\\sqrt{3})\\right]$,\n\nsuch that the slow roll parameters are of the order of unit and the field \nreaches the minimum of the potential at $V(0)= -\\gamma\/(4\\kappa^2)$.\nThe e-foldings number (\\ref{N}) is given by\n\\begin{equation}\nN\\simeq\\frac{3(2-n)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}}{4n^2}\\Big\n\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\n\\sqrt{\\frac{2(2-n)\\gamma}{6}}t_0\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{4n^2N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nAs a consequence, the amplitude of primordial power spectrum and the spectral \nindexes read\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma(2-n)n^2 N^2}{36\\pi^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{1}{N}\\,,\\quad r\\simeq\\frac{12}{n^2 N^2}\\,.\n\\end{equation}\nSince $n_s>1-(\\sqrt{0.11\/12})n\\simeq 1-(0.0957)n$ when $r<0.11\\,,n_s<1$, one \nhas that\n\\begin{equation}\n0.33861$. Since in this case Eq.~(\\ref{alphaalpha}) could lead to $\\alpha<0$ \n(when $n>2$) making the potential unable to reproduce inflation, we may \ngeneralize the potential to the following form\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}\\left(1-\n\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\right)\n-\\frac{\\gamma}{\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\label{V4}\n\\end{equation}\nwhere $\\alpha\\,,\\gamma>0$ and $n>1$.\nNow, in order to recover the Einstein gravity at small curvature, we have to \nput\n\\begin{equation}\n\\alpha=\\frac{\\gamma}{2(n-1)}>0\\,.\n\\end{equation}\nIn the slow roll limit ($\\sigma\\rightarrow-\\infty$) the EOMs read\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{6(n-1)}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \\left(\\frac{\\sqrt{2}\\gamma}{\\sqrt{3\\kappa^2}}\\right)\n\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\n\\end{equation}\nand the analysis of inflation results to be the same of $R^2$-models (with or \nwithout the cosmological constant term).\\\\\n\\\\\nHere, we can give some comments about the results of our investigation. Scalar \ninflationary potentials wich satisfy viable conditions for realistic primordial \nacceleration in the contest of large scalar curvature corrections to General \nRelativity can be classified in two classes. In the first one, the scalar \npotential behaves as \n$V(\\sigma)\\sim\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]\\,,n>0$, and produces \nacceleration with a sufficient amount of inflation only for $n$ very close to \nzero, namely the $F(R)$-model must be close to the Starobinsky one. In the \nsecond class, $V(\\sigma)\\sim\\alpha \n-\\gamma\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, $n>0$, and a quasi de \nSitter solution emerges during inflation. Cosmological data are satisfied if \n$n>1\/3$. This kind of scalar potentials is originated from large scalar \ncurvature corrections of the type $F(R)\\sim c_1 R^2+c_2 R^\\zeta$, with \n$1<\\zeta<5\/3$ when $1\/31\\,,\n\\label{modellino}\n\\end{equation}\nwhere $\\alpha>0$ is a (dimensional) constant parameter and $R_0\\,,\\Lambda$ are \ntwo (cosmological) constants introduced to generalize the model. This class of \nmodels, following the success of quadratic correction, have been often analyzed \nin literature in order to reproduce the dynamics of inflation and\nrecently, in Ref.~\\cite{cinesi}, the cases of polynomial corrections added to \nthe Starobinsky model have been investigated.\nIn what follows, at first we would like to study the\ninflation in the Einstein frame given by (\\ref{modellino}) with $n\\neq2$. We \nwill see that inflation for large magnitude values of the field is realized for \n$n\\lesssim2$. However, other possibilities are allowed by considering \nintermediate values of the field. In this case, the de Sitter solution may \nemerge in the models with $n>2$, but in order to study the exit from inflation \nis necessary to analyze the theory in the Jordan frame, where perturbations \nmake possible an early time acceleration with a sufficient amount of \n$N$-foldings number: that is the aim of the second part of this Section.\n\nLet us start from the potential\n in the scalar field representation of (\\ref{modellino}),\n\\begin{eqnarray}\nV(\\sigma)&=&\\frac{1}{2\\kappa^2}\\left\\{\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\left(1-\\text{e}^{\\sqrt{(2\\kappa^2\/3)}\\sigma}\\right)^{\\frac{n}{n-1}}\\text{e}^{\\frac{(n-2)}{(n-1)}\\sqrt{(2\\kappa^2\/3)}\\sigma}\\left[\\frac{n-1}{n}\\right]\\right.\\nonumber\\\\\n&&\\left.+R_0\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\left(\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}-1\\right)\n-\\Lambda\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}\\right\\}\\,.\n\\end{eqnarray}\nWe immediatly see that only if $n=2$, when $\\sigma\\rightarrow-\\infty$, one gets \n$V(\\sigma)\\sim\\text{const}$ and obtains the de Sitter solution (for $n=2$, \n$\\Lambda=R_0=0$ we recover (\\ref{VStaro}),\n$V(\\sigma)=(\\exp[\\sqrt{2\\kappa^2\/3}\\sigma]-1)^2\/(8\\alpha\\kappa^2)$).\nOn the other hand, even if $10$ very small bounded at the \nbeginning of inflation.\nThe Hubble parameter reads\n\\begin{equation}\nH=\\frac{3(n-1)^2}{(2-n)^2}\\frac{1}{(t_0+t)}\\,,\\quad\\frac{\\ddot \na}{a}=\\frac{3(n-1)^2((n+1)^2-2)}{(n-2)^4}\\frac{1}{(t_0+t)^2}>0\\,,\n\\end{equation}\nand we have an acceleration with decreasing Hubble parameter and curvature\n\\begin{equation}\nR=\\frac{3(n-1)^2\\left(3(n-1)^2-(2-n)\\right)}{(2-n)^4(t_0+t)}\\,.\n\\end{equation}\nThe $N$-foldings number of inflation is given by\n\\begin{equation}\nN\\simeq -\\frac{(n-1)}{(2-n)}\\sqrt{\\frac{3\\kappa^2}{2}}\\sigma\n\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\n-3\\left(\\frac{n-1}{2-n}\\right)^2\\ln\\left[\n\\sqrt{\\frac{2}{3n}}\\frac{(2-n)^2}{6(n-1)^{3\/2}}\\left(\\frac{1}{n\\alpha}\\right)^{\\frac{1}{2(n-1)}}\nt_0\\right]\\,.\n\\end{equation}\nWhen the inflation ends, the field reaches the minimum of the potential. \nWe note that the cosmological constants introduced in the model do not play \nany role at inflationary stage, and change the behaviour of the potential only \nat the end of the inflation.\nIf \n$\\Lambda=R_0=0$, it is easy to see that $V(0)=0$, otherwise the potential \npossesses a minimum before $\\sigma=0$ (in the same way of \\S~\\ref{case}), where \nthe field falls and starts to oscillate.\n\nThe amplitude of primordial power spectrum (\\ref{spectrum}) and the spectral \nindexes (\\ref{indexes}) are\n\\begin{equation}\n\\Delta_{\\mathcal \nR}^2\\simeq\\frac{(n-1)^3\\kappa^2\\text{e}^{\\frac{2}{3}N\\frac{(n-2)^2}{(n-1)^2}}}{16\\pi^2(2-n)^2n}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}\\,,\n\\quad\nn_s\\simeq1-\\frac{8(2-n)}{3(n-1)}\\,,\\quad r\\simeq\\frac{16(2-n)^2}{3(n-1)^2}\\,,\n\\end{equation}\nwhere we have taken into account that $n$ is close to two. In order to make \nthese indexes compatible with (\\ref{data}), we must require\n$ n\\sim 1.8\\,, 1.9$,\nsuch that, as a matter of fact, the quadratic corrections with $n=2$ appear to be \nthe only possibilities of the type (\\ref{modellino}) able to reproduce a viable \ninflation in the Einstein frame. However, it is interesting to extend our \ninvestigation to the Jordan frame.\n\n\\subsection{Inflation in the Jordan frame}\n\nIn the first part of this Section, we have seen how model (\\ref{modellino}) can \nproduce inflation in the Einstein frame representation. We have an early time \nacceleration followed by the end of inflation and the slow roll conditions are \nsatisfied if $n$ is very close to two.\nFor the sake of simplicity, in this Subsection, we \nwill put $\\Lambda=R_0=0$ in (\\ref{modellino}).\nLet us return to the Jordan frame.\n\nThe Friedmann equation for a generic $F(R)$-model read (in vacuum)\n\\begin{equation}\n\\left(R F_R(R)- F(R)\\right)-6H^2 F_R(R)-6H\\dot F_R(R)=0\\,,\\label{eq}\n\\end{equation}\nsuch that for (\\ref{modellino}) one derives\n\\begin{equation}\n\\alpha(R)^{n-1}\\left[R(1-n)\\right]-6H^2\\left[1+\\alpha \nn(R)^{n-1}\\right]=6H\\alpha n\\,(n-1)(R)^{n-2}\\dot R\\,.\\label{First}\n\\end{equation}\nIn the high curvature limit (it means, for large and negative values of the \nscalar field $\\sigma$ in the Einstein frame) we may consider $\\alpha \nn(R)^{n-1}\\gg 1$,\n\\begin{equation}\nH^2\\simeq\\frac{(n-1)}{6n}\\left[R-\\frac{6n\\,H\\,\\dot R}{R}\\right]\\,.\n\\end{equation}\nDuring inflation, the Hubble parameter $H$ evolves slowly and we can consider \nthe anologous of the slow roll parameterers which have the same\nformal dependence of the Einstein frame, namely we require $|\\dot H|\/H^2\\ll \n1$, $|\\ddot H\/(H\\dot H)|\\ll 1$. Thus, we get\n\\begin{equation}\n\\frac{\\dot \nH}{H^2}\\simeq-\\left[\\frac{4-2n}{n+4(n-1)^2}\\right]\\,,\\quad\\frac{\\ddot a}{a}=H^2\n+\\dot H=1-\\frac{4-2n}{n+4(n-1)^2}\\,.\n\\end{equation}\nIt follows that the expansion is accelerated only if\n\\begin{equation}\nn>\\frac{5}{4}=1.25\\,.\n\\end{equation}\nThis value is close to the one given by the left side of (\\ref{nn}). Moreover, \nfor $n>2$, we see that $\\dot H>0$ and the Hubble parameter, and therefore the \ncurvature, grows up with the time and the physics of Standard Model is not \nreached. This result is in agreement with the behaviour of the model discussed \nin the Einstein frame: when $n>2$ and the acceleration emerges in high \ncurvature limit, the field cannot reach a minimum of the potential for small \nvalues and we do not exit from inflation. We could arrive to this conclusion \nalso by looking for Eq.~(\\ref{bimbo}): when $n>2$, $t_0$ has to be very large \nat the beginning of inflation and the scalar field grows up with the time. \nHowever, it does not mean that models with $n>2$ do not produce inflation. In \nscalar field representation we start at very large and negative values of the \nfield (it means, that in Jordan framework we can ignore the $R$-term in the \naction). It may be interesting to see what happen for intermediate values of \nthe field. We analyze the problem in the Jordan frame. From Eq.~(\\ref{eq}) we \nhave the de Sitter condition at $R_0=12H_0^2$, $H_0$ constant, namely\n\\begin{equation}\n2F(R_0)-R_0F_R(R_0)=0\\,,\\label{trace}\n\\end{equation}\nwhich leads in our case,\n\\begin{equation}\nR_0=\\left(\\frac{1}{\\alpha(n-2)}\\right)^{\\frac{1}{n-1}}\\,.\\label{RdS}\n\\end{equation}\nThe model with $n=2$ does not possess an exact de Sitter solution, but if \n$n>2$, we can realize it. Here, we remember that $\\alpha>0$.\nIn the Einstein frame this solution corresponds to\n\\begin{equation}\n\\sigma_{max}=-\\sqrt{\\frac{3}{2\\kappa^2}}\\log\\left[\\frac{2n-2}{n-2}\\right]\\,,\n\\end{equation}\nwhere we have used (\\ref{sigma}). We note that $\\sigma<0$ only if $n>2$ (we are \nassuming $n>1$); otherwise, if $n<2$, this expression becomes imaginary and \nmeaningless.\nThis solution corresponds to a maximum of the potential, because of\n\\begin{equation}\nV'(\\sigma_{max})=0\\,,\\quad \nV''(\\sigma_{max})=\\frac{1}{6}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}\n\\frac{n-2}{n-1}\\left(\\frac{n}{n-2}\\right)^{\\frac{(2-n)}{(n-1)}}>0\\,.\n\\end{equation}\nWhen $n>2$, the potential in Einstein frame has a maximum. The scalar field \nproduces the de Sitter solution, but the field does not evolve with the time \n($\\dot\\sigma=0$) and we do not have a natural exit form inflation. However, by \nmaking use of the perturbative theory, we may study the stability of the model \nin the Jordan frame.\n\nBy perturbating Eq.~(\\ref{eq}) as $R\\rightarrow R+\\delta R$, $|\\delta R|\\ll 1$ \naround the de Sitter solution, we get\n\\begin{equation}\n\\left(\\Box-m^2\\right)\\delta R\\simeq 0\\,,\n\\quad\nm^2=\\frac{1}{3}\\left(\\frac{F'(R)}{F''(R)}-R\n\\right)\\,.\n\\end{equation}\nHere, $m^2$ is the effective mass of the scalaron, namely the new degree of \nfreedom\nintroduced by the modified gravity through $F_R(R)$, which is proportional to \nthe opposite of the inflaton, namely $-\\sigma$. As a consequence, $m^2$ is \nproportional to the opposite of the scalar potential of the inflaton and when \n$m^2<0$ the solution is unstable. In our case, we get on the de Sitter \nsolution,\n\\begin{equation}\nm^2=-(n-2)^{\\frac{n-2}{n-1}}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}<0\\,,\\quad \n22\\,,\n\\end{equation}\nsince it is easy to demonstrate that $x<0$ if $0<16(n-2)\/n$, that is always \ntrue when $20$, the Ricci scalar decreases with the red shift and the physics of \nStandard Model can emerge.\nIn Ref.~\\cite{nostrum} numerical calculations have been executed in different \ninflationary models\nprovided by the specific $R^n$-term with $22$, may represent a valid inflationary scenario. \nIn this case, the curvature is bounded at a specific value at the beginning of \ninflation, and the de Sitter space-time is an exact solution of the model, \nwhich results to be highly unstable. From (\\ref{delta}) we see how extremely \nsmall perturbations in hot universe give the correct $N$-foldings number and \nthe exit from inflation.\nFinally, it is important to note that positive energy density of perturbations \nbrings the curvature to decrease during the early-time acceleration making the \nmodel consistent with the observable evolution of our universe.\n\n\n\\section{Conclusions}\n\n\nThe attention that recently has been paid to modified theories of gravity is \ncaused by the idea of unified description of early-time and late-time cosmic \nacceleration~\\cite{Review-Nojiri-Odintsov}. Moreover,\nthe gravitational action of such a kind of theories may describe quantum \neffects in hot universe scenario, and the last cosmological data seems to be in \nfavour of quadratic corrections to General Relativity during this phase.\n\nIn this paper, we have investigated some feautures of $F(R)$-modified gravity \nmodels for inflationary cosmology, by performing our analysis in the Jordan and \nin the Einstein framework.\n\nAt first, we have studied\ninflation for the class of scalar potentials of the type \n$V(\\sigma)\\sim\\exp[n\\sigma]$, $n$ being a general parameter, in Einstein frame. \nAs a matter of fact, for such a kind of models is possible to reconstruct the \n$F(R)$-gravity theories which correspond to the given potentials.\nSince\nviable inflation must be consistent with the last Planck data, the potentials \nhave been carefully alalyzed, by finding the conditions on the parameters which \nmake possible the early-time acceleration according with $N$-foldings, spectral\nindex and tensor-to-scalar ratio coming from observations.\nWe have derived the form of the $F(R)$-models at the large and small curvature \nlimits, demonstrating that these\nmodels in the Jordan frame correspond to corrections to Einstein gravity which \nemerge only at mass scales larger than the Planck mass.\nThe investigated potentials can be classified in two classes. In the first one, \nthe scalar potential behaves as \n$V(\\sigma)\\sim\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]\\,,n>0$, and produces \nacceleration with a sufficient amount of inflation only for $n$ very close to \nzero, namely the $F(R)$-model must be closed to the Starobinsky one. In the \nsecond class, $V(\\sigma)\\sim\\alpha \n-\\gamma\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, $n>0$, and the quasi de \nSitter solution emerges during viable inflation when $n>1\/3$. This kind of \nscalar potentials is originated from large curvature corrections of the type \n$F(R)\\sim c_1 R^2+c_2 R^\\zeta$, with $1<\\zeta<5\/3$ when $1\/31$.\nIt is interesting to note that $R^2$-term which induces the early-time \ninflation is also responsible for removal of finite-time future \nsingularity in $F(R)$-gravity unifying the inflation with dark energy~\\cite{OdSing}.\n\nIn the second part of the paper, we have studied in detail the specific class \nof models $F(R)=R+(R+R_0)^n$. The analysis in the\nEinstein frame reveals that $n$ must be very close to two in order to realize a \nviable inflation for large and negative values of the scalar field,\nbut other possibilities are allowed\nby starting from intermediate values of the field.\nTo be specific, when $n>2$, the de Sitter solution emerges, but in order to \nstudy the exit from inflation, since in this case the de Sitter space-time is \nan exact solution and the field does not move and exit from inflation in a \nnatural way, is necessary to analyze the theory in the Jordan frame, where \nperturbations make possible an early time acceleration with a sufficient amount \nof inflation. Moreover, we can explicitly demonstrate\nthat the curvature decreases making the model consistent with the historical \nevolution of our universe.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank M. Sami for useful discussions and valuable suggestions.\nThe work by SDO has been supported in part by MINECO (Spain), project \nFIS2010-15640, by AGAUR (Generalitat de Catalunya), contract 2009SGR-994 and by \nproject 2.1839.2011 of MES (Russia).\n\n\n\n\n\\section*{Appendix A}\nLet us consider the equation\n\\begin{equation}\nF_R(R)-\\frac{1}{2}-\\frac{F_R(R)^{1+n}}{2} =-\\frac{R}{4\\gamma(n+2)}\\,,\n\\label{EqA0}\\end{equation}\nwith $n\\,,\\gamma>0$ (it corresponds to (\\ref{pippo}) with (\\ref{alphacond})).\nWe want to find solutions of the above equation as a function of $R$.\nA simple possibility is looking for regular solutions as a power series of $R$, \nnamely\nsetting\n\\begin{equation*}\nF_R(R)=1+c_1R+c_2R^2+c_3R^3+...\\,,\n\\end{equation*}\nwhere the constants $c_{1,2,3...}$ have to be determined. At first, we are \ninterested in solutions where $|c_{1,2,3...}R^{1,2,3...}|\\ll 1$, namely we \nanalyze the limit at small curvature. Thus,\nplugging this Ansatz in the above equation one has\n\\begin{equation*}\n(1+c_1R+c_2R^2+c_3R^3+...)-\\frac{1}{2}-\\frac{1}{2}\\left(1+c_1R+c_2R^2+ \nc_3R^3+... \\right)^{1+n}=\n-\\frac{R}{4\\gamma(n+2)}\\,.\n\\end{equation*}\nPutting\n\\begin{equation*}\nX=+c_1R+c_2R^2+ c_3R^3+... \\,,\n\\end{equation*}\nand assuming $|X|\\ll 1$, the following expansion holds\n\\begin{equation*}\n(1+X)^{1+n}=1+(1+n)X+\\frac{(1+n)n}{2}X^2+\\frac{n(n+1)(n-1)}{3!}X^3+...\\,.\n\\end{equation*}\nAs a consequence, one arrives at the recursive relations\n\\begin{equation*}\nc_1=\\frac{1}{2\\gamma(n-1)(n+2)}\\,,\\quad\n c_2=-\\frac{(1+n)n\\,c_1^2}{2(n-1)}\\,,\\quad\n c_3=-\\frac{(1+n)c_1^3}{6}\\,, ...\\,,\n\\end{equation*}\nfrom which it follows\n\\begin{equation*}\nc_1 R\\propto\\frac{R}{\\gamma}\\,,\\quad\n c_2 R^2\\propto\\frac{R}{\\gamma^2}\\,,\\quad\n c_3 R^3\\propto\\frac{R^3}{\\gamma^3}\\,,...\n\\end{equation*}\n This approximation is valid when $R\\ll\\gamma$.\nOn the other side, when $R\\gg\\gamma$ we can check for the solutions in the \nform\n\\begin{equation*}\nF_R(R)=c_0\\left(\\frac{R}{\\gamma}\\right)^\\zeta\\,,\\quad\\zeta<1\\,.\n\\end{equation*}\nIn such a case from (\\ref{EqA0}) we get\n\\begin{equation*}\nc_0^{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\zeta(1+n)}\\simeq\\frac{1}{4(n+2)}\\left(\\frac{R}{\\gamma}\\right)\\,,\n\\end{equation*}\nand as a consequence\n\\begin{equation*}\nc_0=\\left(\\frac{1}{4(n+2)}\\right)^{\\frac{1}{1+n}}\\,,\\quad\\zeta=\\frac{1}{1+n}<1\\,.\n\\end{equation*}\nNow let us consider the following equation (it corresponds to \\ref{pippo2} \nwith \\ref{alphaalpha}),\n\\begin{equation}\nF_R(R)-F_R(R)^{1-n} =\\frac{R}{2\\gamma(2-n)}\\,,\n\\label{EqA1}\\end{equation}\nwhere $\\gamma>0$ and $0\\alpha_2\\,.\n\\end{equation*}\nSince $0 0$.\nChoose an open set $U \\S X$ such that $x_0 \\in U$ and\nfor all $x \\in U$ we have\n\\[\n\\| a (x) - a (x_0) \\| < \\frac{\\varepsilon}{2}\n\\qquad {\\mbox{and}} \\qquad\n\\| \\alpha_x (a (x_0)) - \\alpha_{x_0} (a (x_0)) \\| < \\frac{\\varepsilon}{2}.\n\\]\nThen, using $\\| \\alpha_x \\| = 1$ for all $x \\in X$,\none sees that $x \\in U$\nimplies $\\| \\alpha_x (a (x)) - \\alpha_{x_0} (a (x_0)) \\| < \\varepsilon$.\n\\end{proof}\n\nFor any group~$G$,\nthere are also algebraic conditions\nrelating the automorphisms $\\alpha_{g, x}$,\ncoming from the requirement that $g \\mapsto \\alpha_g$\nbe a group homomorphism.\nIf $G = {\\mathbb{Z}}$,\nthen we really need only the function $x \\mapsto \\alpha_{1, x}$.\nFor reference,\nwe give the relevant statement as a lemma.\n\n\\begin{lem}\\label{L_4Y12_GetAuto}\nLet $X$ be a locally compact Hausdorff space,\nlet $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a C*-algebra.\nThen there is a one to one correspondence between actions of ${\\mathbb{Z}}$ on\n$C_0 (X, D)$\nthat lie over $h$ and\ncontinuous functions from $X$ to ${\\mathrm{Aut}} (D)$,\ngiven as follows.\n\nFor any function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$,\nthere is an automorphism $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$\ngiven by $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$,\nand this automorphism lies over~$h$.\n\nConversely,\nif $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$ lies over~$h$,\nthen there is a function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$\nand such that $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\n\\end{lem}\n\n\\begin{proof}\nUsing Lemma~\\ref{L_9Y30_MixCont}, this is immediate.\n\\end{proof}\n\nThere is a conflict in the notation in Lemma~\\ref{L_4Y12_GetAuto}:\nif $n \\in {\\mathbb{Z}}$\nthen $\\alpha_n$\nis one of the automorphisms in the action on $C_0 (X, D)$\n(namely $\\alpha_n$),\nwhile if $x \\in X$\nthen $\\alpha_x \\in {\\mathrm{Aut}} (D)$.\nWe use this notation anyway to avoid having more letters.\nTo distinguish the two uses,\ntake $\\alpha$ to be the action\nand write $x \\mapsto \\alpha_x$\nwhen the function from $X$ to ${\\mathrm{Aut}} (D)$ is intended.\n\nIf $D$ is prime,\nthen every action on $C_0 (X, D)$\nlies over an action of $G$ on~$X$.\n\n\\begin{lem}\\label{N_4Y12_IfDSimple}\nLet $X$ be a locally compact Hausdorff space,\nlet $D$ be a prime C*-algebra,\nlet $G$ be a topological group,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X, D))$\nbe an action of $G$ on $C_0 (X, D)$.\nThen there exists an action of $G$ on~$X$\nsuch that $\\alpha$ lies over this action.\n\\end{lem}\n\n\\begin{proof}\nThe action of $G$ on~$X$\nis obtained from the identification\n$X \\cong {\\mathrm{Prim}} ( C_0 (X, D) )$.\n\\end{proof}\n\n\\begin{prp}\\label{P_4Y15_CPSimple}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nthe action is minimal and\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple.\n\\end{prp}\n\n\\begin{proof}\nFor any C*-algebra~$A$,\nlet ${\\widehat{A}}$ be the space of unitary equivalence classes\nof irreducible representations of~$A$,\nwith the hull-kernel topology.\nSince the primitive ideals\nof $C (X, D)$ are exactly the kernels of the point evaluations,\nthere is an obvious map $q \\colon \\CXDHat{X}{D} \\to X$,\nand the open sets in $\\CXDHat{X}{D}$ are exactly\nthe sets $q^{-1} (U)$ for open sets $U \\subset X$.\nIt is now immediate that for every finite set\n$S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in \\CXDHat{X}{D} \\colon\n {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in $\\CXDHat{X}{D}$.\nThat is, the action of $G$ on $\\CXDHat{X}{D}$\nis topologically free in the sense of Definition~1 of~\\cite{AS}.\n\nLet $J \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nbe a nonzero ideal.\nLet\n\\[\n\\pi \\colon C^* \\big( G, \\, C (X, D), \\, \\alpha \\big)\n \\to C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)\n\\]\nbe the quotient map.\nTheorem~1 of~\\cite{AS}\nimplies that $\\pi^{-1} (J)$ has nonzero intersection\nwith the canonical copy of $C (X, D)$\nin $C^* \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nTherefore $J$ has nonzero intersection\nwith the canonical copy of $C (X, D)$\nin $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nSince $D$ is simple,\nthis intersection has the form $C_0 (U, D)$\nfor some nonempty open set $U \\subset X$.\n\nSince the action of $G$ on $X$ is minimal and $X$ is compact,\nthere exist $n \\in {\\mathbb{Z}}_{> 0}$ and $g_1, g_2, \\ldots, g_n \\in {\\mathbb{Z}}_{> 0}$\nsuch that the sets\n$g_1^{-1} U, \\, g_2^{-1} U, \\, \\ldots, g_n^{-1} U$ cover~$X$.\nChoose\n\\[\nf_1, f_2, \\ldots, f_n\n \\in C (X)\n \\subset C (X, D)\n \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)\n\\]\nsuch that ${\\mathrm{supp}} (f_k) \\subset g_k^{-1} U$ for $k = 1, 2, \\ldots, n$\nand $\\sum_{k = 1}^n f_k = 1$.\nFor $k = 1, 2, \\ldots, n$, the functions\n$\\alpha_{g_k} (f_k)$\nare in $C_0 (U) \\subset C_0 (U, D) \\subset J$,\nso\n\\[\n1 = \\sum_{k = 1}^n f_k\n = \\sum_{k = 1}^n u_{g_k}^* \\alpha_{g_k} (f_k) u_{g_k}\n \\in J.\n\\]\nSo $J = C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\n\\end{proof}\n\n\\begin{prp}\\label{P_5418_CPStFin}\nAssume the hypotheses of Proposition~\\ref{P_4Y15_CPSimple},\nand in addition assume that\n$G$ is amenable\nand $D$ has a tracial state.\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nhas a tracial state\nand is stably finite.\n\\end{prp}\n\n\\begin{proof}\nSince $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple},\nit suffices to show that\n$C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nhas a tracial state.\nWe know that the tracial state space\n${\\operatorname{T}} (C (X, D))$ is nonempty,\nsince one can compose a tracial state on~$D$\nwith a point evaluation\n$C (X, D) \\to D$.\nSince $G$ is amenable,\ncombining Theorem 2.2.1 and 3.3.1 of \\cite{Gr}\nshows that $C (X, D)$ has a $G$-invariant tracial state~$\\tau$.\nStandard methods show that the composition of $\\tau$\nwith the conditional expectation\nfrom $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nto $C (X, D)$\nis a tracial state on\n$C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\n\\end{proof}\n\nThe following condition is a technical hypothesis\nwhich we need for the proof of the main large subalgebra result\n(Theorem~\\ref{T_5418_AYLg}).\n\n\\begin{dfn}\\label{D_4Y12_PsPer}\nLet $D$ be a C*-algebra,\nand let $S \\subset {\\mathrm{Aut}} (D)$.\nWe say that $S$ is {\\emph{pseudoperiodic}}\nif for every $a \\in D_{+} \\setminus \\{ 0 \\}$\nthere is $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that for every $\\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\}$\nwe have $b \\precsim \\alpha (a)$.\n\\end{dfn}\n\nThe interpretation of pseudoperiodicity is roughly as follows.\nSuppose $S \\subset {\\mathrm{Aut}} (D)$ is pseudoperidic.\nThen there is no sequence $(\\alpha_n)_{n \\in {\\mathbb{Z}}_{> 0}}$\nin $S$ for which there is a nonzero element\n$\\eta \\in W (D)$\nsuch that the sequence $(\\alpha_{n} (\\eta))_{n \\in {\\mathbb{Z}}_{> 0}}$\nbecomes arbitrarily small in $W (D)$\nin a heuristic sense.\n\nWe give some conditions which imply pseudoperiodicity.\n\n\\begin{lem}\\label{L_4Y12_ApproxInn}\nLet $D$ be a unital C*-algebra.\nThen the set of approximately inner automorphisms of $D$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nIt suffices to prove that\n$a \\precsim \\alpha (a)$ for every approximately inner automorphism\n$\\alpha \\in {\\mathrm{Aut}} (D)$.\nTo see this, let $\\varepsilon > 0$ be arbitrary,\nand use approximate innerness\nto chose a unitary $u \\in D$ such that\n$\\| u \\alpha (a) u^* - a \\| < \\varepsilon$.\n\\end{proof}\n\n\\begin{lem}\\label{Lem1_190201D}\nLet $D$ be a simple C*-algebra.\nLet $S \\S {\\mathrm{Aut}} (D)$\nbe a subset which is compact in the topology\nof pointwise convergence in the norm on~$D$.\nThen $S$ is pseudoperiodic\nin the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nWithout loss of generality{} $\\| a \\| = 1$.\nFor $\\beta \\in S \\cup \\{ {\\mathrm{id}}_D \\}$ set\n\\[\nU_{\\beta}\n = \\left\\{ \\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\} \\colon\n \\| \\alpha (a) - \\beta (a) \\| < \\frac{1}{2} \\right\\}.\n\\]\nBy compactness,\nthere are $\\beta_1, \\beta_2, \\ldots, \\beta_n \\in S \\cup \\{ {\\mathrm{id}}_D \\}$\nsuch that $U_{\\beta_1}, U_{\\beta_2}, \\ldots, U_{\\beta_n}$\ncover $S \\cup \\{ {\\mathrm{id}}_D \\}$.\nSince $\\bigl( \\beta (a) - \\frac{1}{2} \\bigr)_{+} \\neq 0$\nfor all $\\beta \\in {\\mathrm{Aut}} (D)$,\nby Lemma~2.6 of~\\cite{PhLg} there is $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim \\bigl( \\beta_k (a) - \\frac{1}{2} \\bigr)_{+}$\nfor $j = 1, 2, \\ldots, n$.\nLet $\\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\}$.\nChoose $k \\in \\{ 1, 2, \\ldots, n \\}$ such that $\\alpha \\in U_{\\beta_k}$.\nThen $\\| \\beta_k (a) - \\alpha (a) \\| < \\frac{1}{2}$,\nso\n$b \\precsim \\bigl( \\beta_k (a) - \\frac{1}{2} \\bigr)_{+}\n \\precsim \\alpha (a)$.\n\\end{proof}\n\nThe following result will not be used,\nsince large subalgebras are not used in our proofs\nwhen $D$ is purely infinite.\nIt is included as a further example of pseudoperiodicity.\n\n\\begin{lem}\\label{L_9212_PISCA}\nLet $D$ be a purely infinite simple C*-algebra.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nThen $a \\precsim b$ for all $b \\in D_{+} \\setminus \\{ 0 \\}$.\nIn particular,\n$a \\precsim \\alpha (a)$ for all $\\alpha \\in {\\mathrm{Aut}} (D)$.\n\\end{proof}\n\n\\begin{lem}\\label{L_4Y12_TrPrsv}\nLet $D$ be a simple unital C*-algebra{}\nwhich has strict comparison of positive elements.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nIf $D$ is finite dimensional,\nthe conclusion is immediate.\nOtherwise,\nlet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nWithout loss of generality{} $\\| a \\| \\leq 1$.\nThen $\\tau (a) \\leq d_{\\tau} (a)$\nfor all $\\tau \\in {\\operatorname{QT}} (D)$.\nMoreover,\nsince ${\\operatorname{QT}} (D)$ is compact and $D$ is simple,\nthe number\n$\\delta = \\inf_{\\tau \\in {\\operatorname{QT}} (D)} \\tau (a)$\nsatisfies $\\delta > 0$.\nUse Corollary~2.5 of~\\cite{PhLg}\nto find\n$b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n$d_{\\tau} (\\langle b \\rangle ) < \\delta$\nfor all $\\tau \\in {\\operatorname{QT}} (D)$.\nThen\nfor every $\\tau \\in {\\operatorname{QT}} (D)$,\nusing $\\tau \\circ \\alpha \\in {\\operatorname{QT}} (D)$\nat the second step,\nwe have\n\\[\nd_{\\tau} (b)\n < \\delta\n \\leq (\\tau \\circ \\alpha) (a)\n \\leq d_{\\tau} ( \\alpha (a) ).\n\\]\nThe strict comparison hypothesis therefore implies\nthat $b \\precsim_A \\alpha (a)$.\n\\end{proof}\n\n\\begin{dfn}\\label{Def2-181221D}\nLet $A$ be a C*-algebra.\nWe say that the\n{\\emph{order on projections over $A$\nis determined by quasitraces}}\nif whenever $p,q \\in M_{\\infty} (A)$ are projections such that\n$\\tau (p) < \\tau(q) $\nfor all $\\tau \\in \\operatorname{QT} (A)$, then $p \\precsim q$.\n\\end{dfn}\n\n\\begin{lem}\\label{L_5417_SP}\nLet $D$ be a simple unital C*-algebra{}\nwith Property~(SP)\nand such that\nthe order on projections over~$A$ is determined by quasitraces.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nIf $D$ is finite dimensional,\nthe conclusion is immediate.\nOtherwise,\nlet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nChoose a nonzero projection{} $p \\in {\\overline{a D a}}$.\nSince ${\\operatorname{QT}} (D)$ is compact and $D$ is simple,\nthe number\n$\\delta = \\inf_{\\tau \\in {\\operatorname{QT}} (D)} \\tau (p)$\nsatisfies $\\delta > 0$.\nChoose $n \\in {\\mathbb{Z}}_{> 0}$\nsuch that $\\frac{1}{n} < \\delta$.\nUse Lemma~2.3 of~\\cite{PhLg}\nto choose a unitary $u \\in A$\nand a nonzero positive element $b \\in A$\nsuch that the elements\n\\[\nb, \\, u b u^{-1}, \\, u^2 b u^{-2}, \\, \\ldots, \\, u^n b u^{-n}\n\\]\nare pairwise orthogonal.\nChoose a nonzero projection{} $q \\in {\\overline{b D b}}$.\nThen\nfor every $\\tau \\in {\\operatorname{QT}} (D)$,\nusing $\\tau \\circ \\alpha \\in {\\operatorname{QT}} (D)$\nat the third step,\nwe have\n\\[\n\\tau (q)\n \\leq \\frac{1}{n + 1}\n < \\delta\n \\leq (\\tau \\circ \\alpha) (p).\n\\]\nThe strict comparison hypothesis therefore implies\nthat $q$ is Murray-von Neumann equivalent{} to a subprojection of~$\\alpha (p)$.\nIt follows that $q \\precsim_A \\alpha (a)$.\n\\end{proof}\n\n\nThe following definition is intended only for convenience\nin this paper.\n(The condition occurs several times as a hypothesis,\nand is awkward to state.)\n\n\\begin{dfn}\\label{D_4Y16_PsPrGen}\nLet $X$ be a locally compact Hausdorff space,\nlet $G$ be a topological group,\nand let $D$ be a C*-algebra.\nLet $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X, D))$\nbe an action of $G$ on $C_0 (X, D)$\nwhich lies over an action of $G$ on~$X$,\nand let $(g, x) \\mapsto \\alpha_{g, x} \\in {\\mathrm{Aut}} (D)$\nbe as in Definition~\\ref{D_4Y12_OverX}.\nWe say that $\\alpha$ is\n{\\emph{pseudoperiodically generated}}\nif\n\\[\n\\big\\{ \\alpha_{g, x} \\colon {\\mbox{$g \\in G$ and $x \\in X$}} \\big\\}\n\\]\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{dfn}\n\n\\begin{lem}\\label{L_4Y16_WhenPPG}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAs in Lemma~\\ref{L_4Y12_GetAuto},\nlet $( \\alpha_{x} )_{x \\in X}$\nbe the family in ${\\mathrm{Aut}} (D)$\nsuch that $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\nSuppose that the subgroup $H$ of ${\\mathrm{Aut}} (D)$ generated by\n$\\{ \\alpha_x \\colon x \\in X \\}$ is pseudoperiodic.\nThen the action of ${\\mathbb{Z}}$ generated by $\\alpha$\nis pseudoperiodically generated.\n\\end{lem}\n\n\\begin{proof}\nOne checks that if $(n, x) \\mapsto \\alpha_{n, x} \\in {\\mathrm{Aut}} (D)$\nis determined\n(following the notation of Definition~\\ref{D_4Y12_OverX})\nby $\\alpha^n (a) (x) = \\alpha_{n, x} (a (h^{-n} (x))$,\nthen $\\alpha_{n, x} \\in H$ for all $n \\in {\\mathbb{Z}}$ and $x \\in X$.\n\\end{proof}\n\n\\section{The orbit breaking subalgebra for a nonempty set meeting\neach orbit at most once}\\label{Sec_OrbBreak}\n\n\\indent\nLet $h \\colon X \\to X$ be a homeomorphism{}\nof a compact Hausdorff space~$X$,\nand let $D$ be a simple unital C*-algebra.\nFor $Y \\subset X$ closed,\nfollowing Putnam~\\cite{Pt1},\nin Definition~7.3 of~\\cite{PhLg}\nwe defined the $Y$-orbit breaking subalgebra\n$C^* ({\\mathbb{Z}}, X, h)_Y \\subset C^* ({\\mathbb{Z}}, X, h)$.\nHere,\nfor an automorphism $\\alpha \\in {\\mathrm{Aut}} ( C (X, D) )$\nwhich lies over~$h$\nwe define $C^* ({\\mathbb{Z}}, C (X, D), \\alpha)_Y$.\nWe prove that if $X$ is infinite,\n$h$~is minimal,\n$Y$ intersects each orbit at most once,\nand an additional technical condition is satisfied\n(namely, that the action of ${\\mathbb{Z}}$ generated by~$\\alpha$\nis pseudoperiodically generated),\nthen $C^* ({\\mathbb{Z}}, C (X, D), \\alpha)_Y$\nis a large subalgebra of $C^* ({\\mathbb{Z}}, C (X, D), \\alpha)$\nof crossed product type,\nin the sense of Definition~4.9 of~\\cite{PhLg}.\nThis is a generalization\nof Theorem~7.10 of~\\cite{PhLg}.\n\n\n\\begin{ntn}\\label{N_4Y12_CPN}\nLet $G$ be a discrete group,\nlet $A$ be a C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (A)$ be an action of $G$ on~$A$.\nWe identify $A$ with a subalgebra of $C^*_{\\mathrm{r}} (G, A, \\alpha)$\nin the standard way.\nWe let $u_g \\in M (C^*_{\\mathrm{r}} (G, A, \\alpha))$\nbe the standard unitary\ncorresponding to $g \\in G$.\nWhen $G = {\\mathbb{Z}}$, we write just $u$ for the unitary~$u_1$\ncorresponding to the generator $1 \\in {\\mathbb{Z}}$.\nWe let $A [G]$ denote the dense *-subalgebra\nof $C^*_{\\mathrm{r}} (G, A, \\alpha)$\nconsisting of sums $\\sum_{g \\in S} a_g u_g$\nwith $S \\subset G$ finite and $a_g \\in A$ for $g \\in S$.\nWe may always assume $1 \\in S$.\nWe let $E_{\\alpha} \\colon C^*_{\\mathrm{r}} (G, A, \\alpha) \\to A$\ndenote the standard conditional expectation,\ndefined on $A [G]$ by\n$E_{\\alpha} \\left( \\sum_{g \\in S} a_g u_g \\right) = a_1$.\nWhen $\\alpha$ is understood, we just write~$E$.\n\nWhen $G$ acts on a compact Hausdorff space~$X$,\nwe use obvious analogs of this notation for $C^*_{\\mathrm{r}} (G, X)$,\nwith the action as in Notation~\\ref{N_5418_ActCX}.\nIn particular,\nif $G = {\\mathbb{Z}}$ and the action of generated by a homeomorphism{}\n$h \\colon X \\to X$,\nwe have $u f u^* = f \\circ h^{-1}$.\n\\end{ntn}\n\n\\begin{ntn}\\label{N_4Y12_C0UD}\nFor a locally compact Hausdorff space~$X$\nand a C*-algebra~$D$,\nwe identify $C_0 (X, D) = C_0 (X) \\otimes D$\nin the standard way.\nFor an open subset $U \\subset X$,\nwe use the abbreviation\n\\[\nC_0 (U, D)\n = \\big\\{ a \\in C_0 (X, D) \\colon\n {\\mbox{$a (x) = 0$ for all $x \\in X \\setminus U$}} \\big\\}\n \\subset C_0 (X, D).\n\\]\nThis subalgebra is of course canonically isomorphic to\nthe usual algebra $C_0 (U, D)$ when $U$ is considered\nas a locally compact Hausdorff space{} in its own right.\n\\end{ntn}\n\nIn particular,\nif $Y \\subset X$ is closed, then\n\\begin{equation}\\label{Eq_4Y12_C0XYD}\nC_0 (X \\setminus Y, \\, D)\n = \\big\\{ a \\in C_0 (X, D) \\colon\n {\\mbox{$a (x) = 0$ for all $x \\in Y$}} \\big\\}.\n\\end{equation}\n\nThe following definition is the analog of\nDefinition~7.3 of~\\cite{PhLg}.\n\n\\begin{dfn}\\label{D_4Y12_OrbSubalg}\nLet $X$ be a locally compact Hausdorff space,\nlet $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$\nbe an automorphism which lies over~$h$.\nLet $Y \\subset X$ be a nonempty closed subset,\nand, following~(\\ref{Eq_4Y12_C0XYD}), define\n\\[\nC^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)_Y\n = C^* \\big( C_0 (X, D), \\, C_0 (X \\setminus Y, \\, D) u \\big)\n \\subset C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big).\n\\]\nWe call it the {\\emph{$Y$-orbit breaking subalgebra}}\nof $C^* \\big( {\\mathbb{Z}}, C_0 (X, D), \\alpha \\big)$.\n\\end{dfn}\n\nWe show that if $Y$ intersects each orbit of~$h$ at most once,\nand the action of ${\\mathbb{Z}}$ generated by~$\\alpha$\nis pseudoperiodically generated,\nthen $C^* ({\\mathbb{Z}}, C_0 (X, D), \\alpha)_Y$\nis a large subalgebra of $C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)$\nof crossed product type.\n\nThe following lemma is the analog of Proposition~7.5 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_AYStruct}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet\n\\[\nu \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)\n\\qquad {\\mbox{and}} \\qquad\nE_{\\alpha} \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\to C (X, D)\n\\]\nbe as in Notation~\\ref{N_4Y12_CPN}.\nLet $Y \\subset X$ be a nonempty closed subset.\nFor $n \\in {\\mathbb{Z}}$, set\n\\[\nY_n = \\begin{cases}\n \\bigcup_{j = 0}^{n - 1} h^j (Y) & \\hspace{3em} n > 0\n \\\\\n \\varnothing & \\hspace{3em} n = 0\n \\\\\n \\bigcup_{j = 1}^{- n} h^{-j} (Y) & \\hspace{3em} n < 0.\n\\end{cases}\n\\]\nThen\n\\begin{align}\\label{Eq:2816CharOB}\n& C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n\\\\\n& = \\big\\{ a \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\colon\n {\\mbox{$E_{\\alpha} (a u^{-n}) \\in C_0 (X \\setminus Y_n, \\, D)$\n for all $n \\in {\\mathbb{Z}}$}} \\big\\}\n \\notag\n\\end{align}\nand\n\\begin{equation}\\label{Eq:2816Dense}\n{\\overline{C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\cap\nC (X, D) [{\\mathbb{Z}}]}}\n = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{L_4Y12_GetAuto},\nthere exists a function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$\nand which satisfies $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\n\nMost of the proof of Proposition~7.5 of~\\cite{PhLg}\ngoes through with only the obvious changes.\nIn analogy with that proof,\ndefine\n\\[\nB = \\big\\{ a \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\colon\n {\\mbox{$E_{\\alpha} (a u^{-n}) \\in C_0 (X \\setminus Y_n, \\, D)$\n for all $n \\in {\\mathbb{Z}}$}} \\big\\}\n\\]\nand\n\\[\nB_0 = B \\cap C (X) [{\\mathbb{Z}}].\n\\]\nThen $B_0$ is dense in~$B$ by the same reasoning as in~\\cite{PhLg}\n(using Ces\\`{a}ro means and Theorem VIII.2.2 of~\\cite{Dv}).\n\nThe proof of Proposition~7.5 of~\\cite{PhLg}\nshows that\nwhen $0 \\leq m \\leq n$ and also when $0 \\geq m \\geq n$,\nwe have $Y_m \\subset Y_n$,\nthat for $n \\in {\\mathbb{Z}}$,\nwe have\n\\begin{equation}\\label{Eq_4Y14_hnyn}\nh^{-n} (Y_n) = Y_{-n},\n\\end{equation}\nand that for $m, n \\in {\\mathbb{Z}}$,\nwe have $Y_{m + n} \\subset Y_m \\cup h^m (Y_n)$.\nIt then follows,\nas in~\\cite{PhLg},\nthat $B_0$ is a *-algebra,\nand that\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\subset {\\overline{B_0}} = B$.\n\nWe next claim that for all $n \\in {\\mathbb{Z}}$\nand $f \\in C_0 (X \\setminus Y_n, \\, D)$,\nwe have $f u^n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nThe changes to the proof of Proposition~7.5 of~\\cite{PhLg}\nat this point are more substantial.\n\nFor $n = 0$ the claim is trivial.\nLet $n > 0$\nand let $f \\in C_0 (X \\setminus Y_n, \\, D)$.\nDefine $b = (f^* f)^{1 \/ (2 n)}$.\nLet $s \\in C_0 (X \\setminus Y_n, \\, D)''$\nbe the partial isometry in the polar decomposition of~$f$,\nso that $f = s (f^* f)^{1 \/ 2} = s b^n$.\nIt follows from Proposition~1.3 of~\\cite{Cu0}\nthat the element $a_0 = s b$ is in $C_0 (X \\setminus Y_n, \\, D)$.\nMoreover,\n$a_0 (f^* f)^{\\frac{1}{2} - \\frac{1}{2 n}} = f$.\nDefine $a_1 \\in C (X, D)$\nby $a_1 (x) = \\alpha_{h (x)}^{-1} \\big( b (h (x)) \\big)$\nfor $x \\in X$,\nand for $k = 1, 2, \\ldots, n - 2$ inductively define\n$a_{k + 1} \\in C (X, D)$\nby $a_{k + 1} (x) = \\alpha_{h (x)}^{-1} \\big( a_k (h (x)) \\big)$\nfor $x \\in X$.\nThe definition of $Y_n$\nimplies that\n$a_1, a_2, \\ldots, a_{n - 1} \\in C_0 (X \\setminus Y, \\, D)$,\nand we already have\n$a_0 \\in C_0 (X \\setminus Y_n, \\, D) \\S C_0 (X \\setminus Y, \\, D)$.\nTherefore the element\n\\[\na = (a_0 u) (a_1 u) \\cdots (a_{n - 1} u)\n\\]\nis in $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nFor $k = 1, 2, \\ldots, n - 2$ and $x \\in X$, we have\n\\[\n\\alpha (a_{k + 1} ) (x)\n = \\alpha_x (a_{k + 1} (h^{-1} (x))\n = \\alpha_x \\big( \\alpha_{x}^{-1} ( a_k (x) ) \\big)\n = a_k (x),\n\\]\nso $\\alpha (a_{k + 1} ) = a_k$.\nSimilarly, $\\alpha (a_1) = b$.\nNow\n\\begin{align*}\na\n& = a_0 (u a_1 u^{-1}) (u^2 a_2 u^{-2})\n \\cdots \\big( u^{n - 1} a_{n - 1} u^{- (n - 1)} \\big) u^n\n \\\\\n& = a_0 \\alpha (a_1) \\alpha^2 (a_2)\n \\cdots \\alpha^{n - 1} (a_{n - 1}) u^n\n = (s b) b^{n - 1} u^n\n = f u^n.\n\\end{align*}\nSo $f u^n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nFinally,\nsuppose $n < 0$,\nand let $f \\in C_0 (X \\setminus Y_n, \\, D)$.\nIt follows from~(\\ref{Eq_4Y14_hnyn})\nthat $f \\circ h^n \\in C_0 (X \\setminus Y_{- n}, \\, D)$,\nwhence also $(f \\circ h^n)^* \\in C_0 (X \\setminus Y_{- n}, \\, D)$.\nSince $- n > 0$,\nwe therefore get\n\\[\nf u^n = \\big( u^{-n} f^* \\big)^*\n = \\big( (f \\circ h^n)^* u^{- n} \\big)^*\n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y.\n\\]\nThe claim is proved.\n\nIt now follows that\n$B_0 \\subset C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nCombining this result with ${\\overline{B_0}} = B$\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\subset B$,\nwe get $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y = B$.\n\\end{proof}\n\nThe following lemma is the analog of Lemma~7.8 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_78}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nFollowing Notation~\\ref{N_4Y12_CPN},\nlet\n$a \\in C (X, D) [G]\n \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nand let $\\varepsilon > 0$.\nThen there exists $f \\in C (X) \\subset C (X, D)$\nsuch that\n\\[\n0 \\leq f \\leq 1,\n\\qquad\nf a^* a f \\in C (X, D),\n\\qquad {\\mbox{and}} \\qquad\n\\| f a^* a f \\| \\geq \\| E_{\\alpha} (a^* a) \\| - \\varepsilon.\n\\]\n\\end{lem}\n\n\\begin{proof}\nWe follow the proof of Lemma~7.8 of~\\cite{PhLg}.\nSet $b = a^* a$.\nThere are a finite set $T \\subset G$\nwith $1 \\in T$,\nand elements $b_g \\in C (X, D)$ for $g \\in T$,\nsuch that $b = \\sum_{g \\in T} b_g u_g$.\nIf $\\| b_1 \\| \\leq \\varepsilon$ take $f = 0$.\nOtherwise, define\n\\[\nU = \\big\\{ x \\in X \\colon\n \\| b_1 (x) \\| > \\| E (a^* a) \\| - \\varepsilon \\big\\},\n\\]\nwhich is a nonempty open subset of~$X$.\nChoose $V$, $W$, $f$, and $x_0$\nas in the proof of Lemma~7.8 of~\\cite{PhLg}.\nThen,\nas there, $f b f = f b_1 f$.\nMoreover,\n\\[\n\\| f b_1 f \\|\n \\geq f (x_0) \\| b_1 (x_0) \\| f (x_0)\n = \\| b_1 (x_0) \\|\n > \\| E_{\\alpha} (a^* a) \\| - \\varepsilon.\n\\]\nThis completes the proof.\n\\end{proof}\n\nThe following lemma is the analog of Lemma~7.9 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_79}\nLet $G$, $X$, the action of $G$ on $X$,\n$D$, and $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe as in Lemma~\\ref{L_4X21_78}.\nLet $B \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nbe a unital subalgebra\nsuch that,\nfollowing Notation~\\ref{N_4Y12_CPN},\n\\begin{enumerate}\n\\item\\label{L_4X21_79:CXI}\n$C (X, D) \\subset B$.\n\\item\\label{L_4X21_79:FSD}\n$B \\cap C (X, D)[G]$ is dense in~$B$.\n\\end{enumerate}\nLet $a \\in B_{+} \\setminus \\{ 0 \\}$.\nThen there exists $b \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim_{B} a$.\n\\end{lem}\n\n\\begin{proof}\nWe follow the proof of Lemma~7.9 of~\\cite{PhLg},\nusing our Lemma~\\ref{L_4X21_78}\nin place of Lemma~7.8 of~\\cite{PhLg},\nexcept that $c \\in B \\cap C (X, D)[G]$.\nThe element $(f c^* c f - 2 \\varepsilon)_{+}$\nwe obtain now satisfies\n$(f c^* c f - 2 \\varepsilon)_{+} \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nand $(f c^* c f - 2 \\varepsilon)_{+} \\precsim_{B} a$.\n\\end{proof}\n\nIn Lemma~\\ref{L_4X21_79},\nwe really want to have $b \\in C (X)_{+} \\setminus \\{ 0 \\}$.\nWhen $G = {\\mathbb{Z}}$ and under the pseudoperiodicity hypothesis\nof \\Def{D_4Y16_PsPrGen}, this is possible.\n\n\\begin{lem}\\label{L_4Y11_Comp}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact set\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nSet $B = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nThen for every $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nthere exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset B$\nsuch that $f \\precsim_B a$.\n\\end{lem}\n\n\\begin{proof}\nUse Kirchberg's Slice Lemma (Lemma 4.1.9 of~\\cite{Rrd})\nto find $g_0 \\in C (X)_{+} \\setminus \\{ 0 \\}$\nand $d_0 \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_5417_Zero}\ng_0 \\otimes d_0 \\precsim_{C (X, D)} a.\n\\end{equation}\nAs in Definition~\\ref{D_4Y12_OverX},\nlet $(m, x) \\mapsto \\alpha_{m, x}$\nbe the function ${\\mathbb{Z}} \\times X \\to {\\mathrm{Aut}} (D)$\nsuch that $\\alpha^m (a) (x) = \\alpha_{m, x} (a (h^{-m} (x))$\nfor $m \\in {\\mathbb{Z}}$, $a \\in C_0 (X, D)$, and $x \\in X$.\nSince the action generated by $\\alpha$\nis pseudoperiodically generated,\nthere exists $d_1\\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_5418_PsPer}\n\\alpha_{m, x} (d_1) \\precsim d_0\n\\end{equation}\nfor all $x \\in X$ and $m \\in {\\mathbb{Z}}$.\nWithout loss of generality{} $\\| d_1 \\| = 1$.\nSet $d = \\big( d_1 - \\tfrac{1}{2} \\big)_{+}$.\nUse Corollary 1.14 of~\\cite{PhLg}\nto find $k \\in {\\mathbb{Z}}_{> 0}$ and $w_1, w_2, \\ldots, w_k \\in D$\nsuch that\n\\begin{equation}\\label{Eq_5418_Sum1}\n\\sum_{j = 1}^k w_j d w_j^* = 1.\n\\end{equation}\n\nThe set $X \\setminus Y$ is dense in~$X$,\nso there is $x_0 \\in X \\setminus Y$ such that $g_0 (x_0) \\neq 0$.\nChoose $g_1 \\in C (X)_{+}$ such that $g_1 (x_0) = 1$\nand $g_1 |_Y = 0$.\nThen\n\\begin{equation}\\label{Eq_5417_One}\ng_1 g_0 \\otimes d_0\n \\precsim_{C (X, D)} g_0 \\otimes d_0.\n\\end{equation}\nSet\n$U_0 = \\big\\{ x \\in X \\colon (g_1 g_0) (x) \\neq 0 \\big\\}$.\nChoose a nonempty open set $U \\subset X$\nsuch that ${\\overline{U}} \\subset U_0$.\nSet\n\\begin{equation}\\label{Eq_9Z01_NewLabel}\n\\rho = \\inf_{x \\in {\\overline{U}}} (g_0 g_1) (x).\n\\end{equation}\nThen $\\rho > 0$.\nThe set\n\\[\nX \\setminus \\bigcup_{n \\in {\\mathbb{Z}}} h^n (Y)\n = \\bigcap_{n \\in {\\mathbb{Z}}} h^n (X \\setminus Y)\n\\]\nis dense by the Baire Category Theorem.\nSo we can choose\n$y_0 \\in U \\cap \\left( X \\setminus\n\\bigcup_{n \\in {\\mathbb{Z}}} h^n (Y) \\right)$.\nThe forward orbits $\\{ h^n (y_0) \\colon n \\geq N \\}$\nof $y_0$ are all dense,\nso there exist $n_1, n_2, \\ldots, n_k \\in {\\mathbb{Z}}_{\\geq 0}$\nwith $0 = n_1 < n_2 < \\cdots< n_k$\nand $h^{n_j} (y_0) \\in U$ for $j = 1, 2, \\ldots, k$.\n\nChoose an open set $V$ containing $y_0$ which is so small that\nthe following hold:\n\\begin{enumerate}\n\\item\\label{L_4Y11_Comp_Cont}\n${\\overline{V}} \\subset X \\setminus\n\\bigcup_{m = 0}^{n_k} h^{- m} (Y)$.\n\\item\\label{L_4Y11_Comp_InU}\n$h^{n_j} ( {\\overline{V}} ) \\subset U$\nfor $j = 1, 2, \\ldots, k$.\n\\item\\label{L_4Y11_Comp_Disj}\nThe sets\n${\\overline{V}}, \\, h ( {\\overline{V}} ), \\, \\ldots, \\,\n h^{n_k} ( {\\overline{V}} )$\nare disjoint.\n\\item\\label{L_4Y11_Comp_Small}\nFor $y \\in V$ and $m = 0, 1, \\ldots, n_k$,\nwe have\n\\[\n\\bigl\\| \\alpha_{m, h^m (y)} (d_1) - \\alpha_{m, h^m (y_0)} (d_1) \\bigr\\|\n< \\frac{1}{4}.\n\\]\n\\setcounter{TmpEnumi}{\\value{enumi}}\n\\end{enumerate}\nChoose $f_0, f \\in C (X)_{+}$\nsuch that:\n\\begin{enumerate}\n\\setcounter{enumi}{\\value{TmpEnumi}}\n\\item\\label{L_4Y11_Comp_1}\n$\\| f_0 \\|, \\, \\| f \\| \\leq 1$.\n\\item\\label{L_4Y11_Comp_Supp}\n${\\mathrm{supp}} (f_0) \\subset V$.\n\\item\\label{L_4Y11_Comp_b0b}\n$f_0 f = f$.\n\\item\\label{L_4Y11_Comp_Aty0}\n$f (y_0) = 1$.\n\\end{enumerate}\nFor $m = 0, 1, \\ldots, n_k - 1$,\ndefine $c_m \\in C (X, D)$\nby $c_m (x) = f ( h^{- m} (x)) \\alpha_{m, x} (d)$\nfor $x \\in X$.\nThus\n\\begin{equation}\\label{Eq_5417_cm}\nc_m = \\alpha^m (f \\otimes d).\n\\end{equation}\nAlso set\n\\begin{equation}\\label{Eq_5418_DefBt}\n\\beta_m = \\alpha_{m, h^m (y_0)}.\n\\end{equation}\n\nWe claim that for $m = 0, 1, \\ldots, n_k$\nwe have\n\\[\nc_m \\precsim_{C (X, D)} (f \\circ h^{-m}) \\otimes \\beta_m (d_1).\n\\]\n\nTo prove the claim, let $\\varepsilon > 0$.\nSet $L = h^{m} ( {\\mathrm{supp}} (f_0) )$.\nDefine $b \\in C (L, D)$\nby $b (x) = \\alpha_{m, x} (d_1)$ for $x \\in L$.\nSince $L \\subset h^m (V)$,\ncondition~(\\ref{L_4Y11_Comp_Small})\nimplies that\nfor $x \\in L$\nwe have $\\| \\alpha_{m, x} (d_1) - \\beta_m (d_1) \\| < \\frac{1}{2}$.\nIn $C (L, D)$ we therefore get\n$\\| b - 1 \\otimes \\beta_m (d_1) \\|\n \\leq \\frac{1}{4} < \\frac{1}{2}$.\nSo\n\\[\n\\bigl( b - \\tfrac{1}{2} \\bigr)_{+}\n \\precsim_{C (L, D)} 1 \\otimes \\beta_m (d_1).\n\\]\nTherefore there exists $v_0 \\in C (L, D)$\nsuch that\n\\[\n\\big\\| v_0^* ( 1 \\otimes \\beta_m (d_1) ) v_0\n - \\bigl( b - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n < \\frac{\\varepsilon}{2}.\n\\]\nDefine $v, a \\in C (X, D)$ by\n\\[\nv (x) = \\begin{cases}\n f_0 (h^{- m} (x)) v_0 (x) & x \\in L\n \\\\\n 0 & x \\not\\in L\n\\end{cases}\n\\qquad {\\mbox{and}} \\qquad\na = v^* \\bigl[ (f \\circ h^{-m}) \\otimes \\beta_m (d_1) \\bigr] v.\n\\]\nWe show that $\\| a - c_m \\| < \\varepsilon$.\n\nFor $x \\in X \\setminus L$,\nwe have $a (x) = c_m (x) = 0$.\nFor $x \\in L$,\nusing $f_0 f = f$ at the second step\nand using $d = \\bigl( d_1 - \\tfrac{1}{2} \\bigr)_{+}$\nand the definition of~$b$ at the third step,\nwe get\n\\begin{align*}\n& \\| a (x) - c_m (x) \\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n = \\big\\| f_0 (h^{- m} (x))^2 f (h^{- m} (x)) v_0 (x)^*\n \\beta_m (d_1) v_0 (x)\n - f (h^{- m} (x)) \\alpha_{m, x} (d) \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n = \\big\\| f (h^{- m} (x))\n [ v_0 (x)^* \\beta_m (d_1) v_0 (x) - \\alpha_{m, x} (d) ] \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n = f (h^{- m} (x))\n \\big\\| v_0 (x)^* \\beta_m (d_1) v_0 (x)\n - \\bigl( b (x) - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n \\leq f (h^{- m} (x))\n \\big\\| v_0^* ( 1 \\otimes \\beta_m (d_1) ) v_0\n - \\bigl( b - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n < \\frac{\\varepsilon}{2}.\n\\end{align*}\nTaking the supremum over $x \\in X$ gives\n$\\| a - c_m \\| \\leq \\frac{\\varepsilon}{2} < \\varepsilon$.\nThus\n\\[\n\\big\\| v^* [ (f \\circ h^{-m}) \\otimes \\beta_m (d_1) ] v - c_m \\big\\|\n < \\varepsilon.\n\\]\nSince $\\varepsilon > 0$ is arbitrary,\nthe claim follows.\n\nFor $m = 0, 1, \\ldots, n_k$,\nthe functions $f \\circ h^{-m}$ are orthogonal\nsince the sets $h^m ( {\\overline{V}} )$ are disjoint.\nThe claim therefore implies that\n\\begin{equation}\\label{Eq_5417_Two}\n\\sum_{j = 1}^k c_{n_j}\n \\precsim_{C (X, D)}\n \\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1).\n\\end{equation}\n\nWe now claim that\n\\begin{equation}\\label{Eq_5417_Three}\nf \\otimes 1 \\precsim_{B} \\sum_{j = 1}^k c_{n_j}.\n\\end{equation}\nTo prove this claim,\nfor $j = 1, 2, \\ldots, k$\ndefine\n\\[\nv_j = [(f \\circ h^{-n_j}) \\otimes 1] u^{n_j} (1 \\otimes w_j)^*\n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr).\n\\]\nCombining (\\ref{L_4Y11_Comp_Supp}) and~(\\ref{L_4Y11_Comp_Cont}),\nwe see that $f_0$ vanishes in particular on the sets\n\\[\nh^{-1} ( Y ), \\, h^{-2} ( Y ), \\, \\ldots, \\, h^{- n_j} ( Y ),\n\\]\nwhence $f_0 \\circ h^{- n_j}$ vanishes on the sets\n\\[\nY, \\, h ( Y ), \\, \\ldots, \\, h^{n_j - 1} ( Y ).\n\\]\nSo $v_j \\in B$ by\n\\Lem{L_4X21_AYStruct}.\nUsing (\\ref{Eq_5417_cm}) at the first step\nand $f_0 f = f$ at the last step,\nwe calculate:\n\\begin{align*}\nv_j^* c_{n_j} v_j\n& = (1 \\otimes w_j) u^{- n_j} [(f \\circ h^{-n_j}) \\otimes 1]\n \\alpha^{n_j} (f \\otimes d) [(f \\circ h^{-n_j}) \\otimes 1]\n u^{n_j} (1 \\otimes w_j)^*\n\\\\\n& = (1 \\otimes w_j) (f_0 \\otimes 1) (f \\otimes d)\n (f_0 \\otimes 1) (1 \\otimes w_j^*)\n = f \\otimes w_j d w_j^*.\n\\end{align*}\nWe apply (\\ref{Eq_5418_Sum1}) at the first step\nand use orthogonality\nof $c_0, c_1, \\ldots, c_{n_k}$\nand Lemma 1.4(12) of~\\cite{PhLg} at the second step,\nto get\n\\[\nf \\otimes 1\n = \\sum_{j = 1}^k f \\otimes w_j d w_j^*\n \\precsim_B \\sum_{j = 1}^k c_{n_j}.\n\\]\nThis proves the claim~(\\ref{Eq_5417_Three}).\n\nWe next claim that\n\\begin{equation}\\label{Eq_5417_Four}\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)} g_0 g_1 \\otimes d.\n\\end{equation}\nTo prove this,\ncombine (\\ref{Eq_5418_PsPer}), (\\ref{Eq_5418_DefBt}),\nand Lemma 1.11 of~\\cite{PhLg}\nto get\n\\[\n(f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)} (f \\circ h^{-n_j}) \\otimes d_0\n\\]\nfor $j = 1, 2, \\ldots, k$.\nSince the functions\n$f \\circ h^{-n_j}$ are orthogonal,\nit follows that\n\\[\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)}\n \\left( \\sum_{j = 1}^k (f \\circ h^{-n_j}) \\right) \\otimes d_0.\n\\]\nUsing orthogonality\nof the functions\n$f \\circ h^{-n_j}$ again,\ntogether with\n\\[\n{\\mathrm{supp}} (f \\circ h^{-n_j}) \\subset h^{n_j} ( {\\overline{V}} )\n \\subset U\n\\qquad {\\mbox{and}} \\qquad\n0 \\leq f \\circ h^{-n_j} \\leq 1\n\\]\nfor $j = 1, 2, \\ldots, k$,\nwe see that\n$0 \\leq \\sum_{j = 1}^k f \\circ h^{-n_j} \\leq 1$\nand,\nby~(\\ref{L_4Y11_Comp_InU}), that this sum is supported in~$U$.\nSince $g_0 g_1 \\geq \\rho$ on ${\\overline{U}}$\nby~(\\ref{Eq_9Z01_NewLabel}),\nwe get the first step in the following computation;\nthe second step is clear:\n\\[\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes d_0\n \\leq \\frac{1}{\\rho} ( g_0 g_1 \\otimes d_0 )\n \\sim_{C (X, D)} g_0 g_1 \\otimes d_0.\n\\]\nThe claim is proved.\n\nNow combine\n(\\ref{Eq_5417_Zero}),\n(\\ref{Eq_5417_One}),\n(\\ref{Eq_5417_Four}),\n(\\ref{Eq_5417_Two}),\nand\n(\\ref{Eq_5417_Three})\nto get $f \\otimes 1 \\precsim_{B} a$.\n\\end{proof}\n\n\\begin{cor}\\label{C_5418_CompCX}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact set\n(possibly empty)\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nSet $B = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nLet $a \\in B_{+} \\setminus \\{ 0 \\}$.\nThen there exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset B$\nsuch that $f \\precsim_B a$.\n\\end{cor}\n\n\\begin{proof}\n\\Lem{L_4X21_AYStruct}\nimplies that $B$ satisfies the hypotheses\nof \\Lem{L_4X21_79}\nwith $G = {\\mathbb{Z}}$.\nTherefore,\nby \\Lem{L_4X21_79},\nthere exists $b \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim_{B} a$.\nNow \\Lem{L_4Y11_Comp}\nprovides $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\precsim_B a$.\n\\end{proof}\n\nWe can now prove the analog of Theorem~7.10 of~\\cite{PhLg}.\n\n\\begin{thm}\\label{T_5418_AYLg}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra{} which has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type\nin the sense of Definition~4.9 of~\\cite{PhLg}.\n\\end{thm}\n\n\\begin{proof}\nWe verify the hypotheses of\nProposition~4.11 of~\\cite{PhLg}.\nWe follow Notation~\\ref{N_4Y12_CPN}.\nSet\n\\[\nA = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr),\n\\qquad\nB = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y,\n\\]\n\\[\nC = C (X, D),\n\\qquad {\\mbox{and}} \\qquad\nG = \\{ u \\}.\n\\]\n\nThe algebra $A$ is simple\nby Proposition~\\ref{P_4Y15_CPSimple}\nand finite by Proposition~\\ref{P_5418_CPStFin}.\nIn particular,\ncondition~(1)\nin Proposition 4.11 of~\\cite{PhLg} holds.\n\nWe next verify condition~(2)\nin Proposition 4.11 of~\\cite{PhLg}.\nAll parts are obvious except~(2d).\nSo let $a \\in A_{+} \\setminus \\{ 0 \\}$\nand $b \\in B_{+} \\setminus \\{ 0 \\}$.\nApply Corollary~\\ref{C_5418_CompCX} twice,\nthe first time with $Y = \\varnothing$ and $a$ as given\nand the second time with $Y$ as given\nand with $b$ in place of~$a$.\nWe get $a_0, b_0 \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $a_0 \\precsim_{A} a$ and $b_0 \\precsim_{B} b$.\n\nWe can now argue as in the corresponding part of the\nproof of Theorem~7.10 of~\\cite{PhLg}\nto find $f \\in C (X)$\nsuch that\n\\[\nf \\precsim_{C (X)} a_0 \\precsim_{A} a\n\\qquad {\\mbox{and}} \\qquad\nf \\precsim_{B} b_0 \\precsim_{B} b,\n\\]\ncompleting the proof of condition~(2d).\nAlternatively,\n$a_0, b_0\n \\in C^* ({\\mathbb{Z}}, X, h)_Y\n \\subset C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$,\nand $C^* ({\\mathbb{Z}}, X, h)_Y$ is a large subalgebra of $C^* ( {\\mathbb{Z}}, X, h)$\nof crossed product type\nby Theorem~7.10 of~\\cite{PhLg},\nso the existence of $f$\nfollows from condition~(2d)\nin Proposition 4.11 of~\\cite{PhLg}.\n\nWe next verify condition~(3)\nin Proposition 4.11 of~\\cite{PhLg}.\nLet $m \\in {\\mathbb{Z}}_{> 0}$,\nlet $a_1, a_2, \\ldots, a_m \\in A$,\nlet $\\varepsilon > 0$,\nand let $b \\in B_{+} \\setminus \\{ 0 \\}$.\nWe follow the corresponding part of the\nproof of Theorem~7.10 of~\\cite{PhLg}.\nChoose $c_1, c_2, \\ldots, c_m \\in C (X, D) [{\\mathbb{Z}}]$\nsuch that $\\| c_j - a_j \\| < \\varepsilon$\nfor $j = 1, 2, \\ldots, m$.\n(This estimate is condition~(3b).)\nChoose $N \\in {\\mathbb{Z}}_{> 0}$\nsuch that for $j = 1, 2, \\ldots, m$\nthere are $c_{j, l} \\in C (X, D)$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$\nwith\n\\[\nc_j = \\sum_{l = -N}^N c_{j, l} u^l.\n\\]\n\nApply Corollary~\\ref{C_5418_CompCX} to~$B$,\nto find $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\precsim_{B} b$.\nSet $U = \\{ x \\in X \\colon f (x) \\neq 0 \\}$,\nand choose nonempty disjoint open sets\n$U_l \\subset U$ for $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$.\nFor each such~$l$,\nuse Lemma~7.7 of~\\cite{PhLg}\nto choose $f_l, r_l \\in C (X)_{+}$\nsuch that $r_l (x) = 1$ for all $x \\in h^l (Y)$,\nsuch that $0 \\leq r_l \\leq 1$,\nsuch that ${\\mathrm{supp}} (f_l) \\subset U_l$,\nand such that $r_l \\precsim_{C^* ({\\mathbb{Z}}, X, h)_Y} f_l$.\nThen also $r_l \\precsim_{B} f_l$.\n\nChoose an open set $W$ containing~$Y$\nsuch that\n\\[\nh^{-N} (W), \\, h^{- N + 1} (W),\n \\, \\ldots, \\, h^{N - 1} (W), \\, h^{N} (W)\n\\]\nare disjoint,\nand choose $r \\in C (X)$ such that $0 \\leq r \\leq 1$,\n$r (x) = 1$ for all $x \\in Y$,\nand ${\\mathrm{supp}} (r) \\subset W$.\nSet\n\\[\ng_0 = r \\cdot \\prod_{l = - N}^{N} r_l \\circ h^{l}.\n\\]\nSet $g_l = g_0 \\circ h^{- l}$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$.\nThen $0 \\leq g_l \\leq r_l \\leq 1$.\nSet $g = \\sum_{l = - N}^{N} g_l$.\nThe supports of the functions $g_l$ are disjoint,\nso $0 \\leq g \\leq 1$.\nThis is condition~(3a)\nin Proposition 4.11 of~\\cite{PhLg}.\nThe proof of Theorem~7.10 of~\\cite{PhLg}\nshows that\n$g \\precsim_{C^* ({\\mathbb{Z}}, X, h)_Y} f \\precsim_B b$.\nSince $C^* ({\\mathbb{Z}}, X, h)_Y \\subset B$,\nit follows that $g \\precsim_B b$.\nThis is condition~(3d)\nin Proposition 4.11 of~\\cite{PhLg}.\n\nIt remains to verify condition~(3c)\nin Proposition 4.11 of~\\cite{PhLg}.\nThis is done the same way as\nthe corresponding part of the proof of Theorem~7.10 of~\\cite{PhLg},\nexcept using \\Lem{L_4X21_AYStruct}\nin place of Proposition~7.5 of~\\cite{PhLg}.\n\\end{proof}\n\nThe following corollary is the analog of Corollary~7.11\nof~\\cite{PhLg}.\n\n\\begin{cor}\\label{C_5418_AYStabLg}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal\nhomeomorphism, let $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a stably large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin the sense of Definition~5.1 of~\\cite{PhLg}.\n\\end{cor}\n\n\\begin{proof}\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is stably finite\n(by Proposition~\\ref{P_5418_CPStFin}),\nwe can combine Theorem~\\ref{T_5418_AYLg},\nProposition~4.10 of~\\cite{PhLg},\nand Corollary~5.8 of~\\cite{PhLg}.\n\\end{proof}\n\n\\begin{cor}\\label{AyCentLarge}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal\nhomeomorphism, let $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin the sense of Definition~3.2 of~\\cite{ArPh}.\n\\end{cor}\n\n\\begin{proof}\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is stably finite\n(by Proposition~\\ref{P_5418_CPStFin}),\nwe can combine Theorem~\\ref{T_5418_AYLg}\nand Theorem~4.6 of~\\cite{ArPh}.\n\\end{proof}\n\nWe conclude by giving some conditions on $D$ and~$\\alpha$ which\nguarantee the hypotheses of Corollary~\\ref{AyCentLarge}.\nThese are more natural to consider than the awkward\npseudoperiodicity hypothesis.\n\n\\begin{cor}\\label{AyCentLargeConds}\nLet $X$ be a compact metric space,\nlet $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $Y \\subset X$ be a compact subset\nsuch that $h^{n} (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\set{0}$,\nlet $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $x \\mapsto \\alpha_{x}$\nbe the corresponding map from $X$ to ${\\mathrm{Aut}} (D)$,\nas in Lemma~\\ref{L_4Y12_GetAuto}.\nAssume one of the following conditions holds.\n\\begin{enumerate}\n\\item\\label{9212_AyCentLargeConds_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9212_AyCentLargeConds_StrCmp}\n$D$ has strict comparison of positive elements.\n\\item\\label{9212_AyCentLargeConds_SP}\n$D$ has property (SP) and the order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9212_AyCentLargeConds_CptGen}\nThe set\n$\\set{ \\alpha_{x} \\colon x \\in X}$\nis contained in a subgroup of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nThen\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\n\\end{cor}\n\n\\begin{proof}\nWe claim that, under any of the conditions\n(\\ref{9212_AyCentLargeConds_AppInn}),\n(\\ref{9212_AyCentLargeConds_StrCmp}),\n(\\ref{9212_AyCentLargeConds_SP}),\nor (\\ref{9212_AyCentLargeConds_CptGen}),\nthe set $\\set{ \\alpha_{x} \\colon x \\in X}$\nis contained in a pseudoperiodic subgroup of ${\\mathrm{Aut}} (D)$.\nFor~(\\ref{9212_AyCentLargeConds_AppInn}),\nthis follows from Lemma~\\ref{L_4Y12_ApproxInn};\nfor~(\\ref{9212_AyCentLargeConds_StrCmp}),\nfrom Lemma~\\ref{L_4Y12_TrPrsv};\nfor~(\\ref{9212_AyCentLargeConds_SP}),\nfrom Lemma~\\ref{L_5417_SP};\nand for~(\\ref{9212_AyCentLargeConds_CptGen}),\nfrom Lemma~\\ref{Lem1_190201D}.\nNow apply Lemma~\\ref{L_4Y16_WhenPPG}\nto see that\nthe hypotheses of Corollary~\\ref{AyCentLarge} are satisfied.\n\\end{proof}\n\nThe case in which $D$ is purely infinite and simple is\nmuch easier.\nWe give a direct proof, not depending on large subalgebras,\nthat $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis purely infinite simple for any discrete group~$G$.\nIt is still true,\nand will be proved below (Proposition \\ref{T_9921_PICase}),\nthat, under the other hypotheses of Theorem~\\ref{T_5418_AYLg}\nthe subalgebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type.\nThis fact seems potentially useful,\nbut does not help with the analysis of any of the examples\nin Section~\\ref{Sec_InfRed}.\n\n\\begin{thm}\\label{T_0101_PICase}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nthe action is minimal and\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis purely infinite simple.\n\\end{thm}\n\nWe saw in Proposition~\\ref{P_4Y15_CPSimple}\nthat $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple,\nbut we won't use that in this proof.\n\n\\begin{proof}[Proof of Theorem~\\ref{T_0101_PICase}]\nFor convenience of notation,\nset $A = C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nWe also freely identify $C (X, D)$ with $C (X) \\otimes D$.\nWe prove the result by showing that $1_A \\precsim a$\nfor all $a \\in A_{+} \\setminus \\{ 0 \\}$.\nBy Lemma~\\ref{L_4X21_79}\n(taking $B$ there to be~$A$),\nwe can assume that $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$.\n\nUse Kirchberg's Slice Lemma (Lemma 4.1.9 of~\\cite{Rrd})\nto find $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nand $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_0201_Slice}\nf \\otimes b \\precsim_{C (X, D)} a.\n\\end{equation}\nWithout loss of generality{} $\\| f \\| = 1$.\nSince $D$ is purely infinite and simple,\nwe have $1_D \\precsim_{D} b$, and so it follows that\n\\begin{equation}\\label{Eq_0201_InD}\nf \\otimes 1_D \\precsim_{D} f \\otimes b.\n\\end{equation}\nSet $U = \\bigl\\{ x \\in X \\colon f (x) > \\frac{1}{2} \\bigr\\}$.\nBy minimality of the action,\nthe sets $g U$ for $g \\in G$ cover~$X$.\nSo there are $n \\in {\\mathbb{Z}}_{> 0}$ and $g_1, g_2, \\ldots, g_n \\in G$\nsuch that the sets $g_1 U, g_2 U, \\ldots, g_n U$ cover~$X$.\nThe function $x \\mapsto \\sum_{k = 1}^n f (g_k^{-1} x)$\nis strictly positive on~$X$.\nUsing this fact at the first step, pure infiniteness of~$D$\nat the second last step,\nand (\\ref{Eq_0201_InD}) and~(\\ref{Eq_0201_Slice})\nat the last step,\nwe get\n\\begin{align*}\n1_A\n& \\precsim_{C (X)} \\sum_{k = 1}^n \\alpha_{g_k} (f \\otimes 1_D)\n\\\\\n& \\precsim_{C (X)} {\\mathrm{diag}} \\bigl( \\alpha_{g_1} (f \\otimes 1_D), \\,\n \\alpha_{g_2} (f \\otimes 1_D), \\, \\ldots, \\,\n \\alpha_{g_n} (f \\otimes 1_D) \\bigr)\n\\\\\n& = {\\mathrm{diag}} \\bigl( u_{g_1} (f \\otimes 1_D) u_{g_1}^*, \\,\n u_{g_2} (f \\otimes 1_D) u_{g_2}^*, \\, \\ldots, \\,\n u_{g_n} (f \\otimes 1_D) u_{g_n}^* \\bigr)\n\\\\\n& \\sim_{A} {\\mathrm{diag}} \\bigl( f \\otimes 1_D, \\,\n f \\otimes 1_D, \\, \\ldots, \\,\n f \\otimes 1_D \\bigr)\n = f \\otimes 1_{M_n (D)}\n \\precsim_{C (X, D)} f \\otimes 1_D\n \\precsim_{A} a.\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\nAs promised,\nwe now give a result on large subalgebras when $G = {\\mathbb{Z}}$.\nWe use two lemmas,\nthe first of which has a similar proof to that of\nTheorem~\\ref{T_0101_PICase}.\n\n\\begin{lem}\\label{L_0101_SubalgPI}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis purely infinite and simple.\n\\end{lem}\n\n\\begin{proof}\nFor convenience of notation,\nset $B = C^* \\big( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\big)_Y$.\nWe also freely identify $C (X, D)$ with $C (X) \\otimes D$.\nWe prove the result by showing that $1_A \\precsim a$\nfor all $a \\in A_{+} \\setminus \\{ 0 \\}$.\nBy Lemma~\\ref{L_4X21_79}\nand Lemma~\\ref{L_4X21_AYStruct},\nwe can assume that $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$.\nUsing Kirchberg's Slice Lemma\nand pure infiniteness of~$D$\nas in the proof of Theorem~\\ref{T_0101_PICase},\nthere is $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\otimes 1_D \\precsim_{C (X, D)} a$.\n\nWe now claim that for every $x \\in X$ there is $l_x \\in C (X)_{+}$\nsuch that $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$\nand $l_x (x) > \\frac{1}{2}$.\nGiven this,\nthe proof is finished as in the last computation\nin the proof of\nTheorem~\\ref{T_0101_PICase}.\n\nTo prove the claim,\nset $U = \\bigl\\{ x \\in X \\colon f (x) > \\frac{1}{2} \\bigr\\}$.\nIf $x \\in U$,\ntake $l_x = f$.\n\nSuppose next that $x \\not\\in U$\nand $x \\not\\in \\bigcup_{m = 0}^{\\infty} h^m (Y)$.\nBy minimality of~$h$,\nthere is $n \\in {\\mathbb{Z}}_{> 0}$ such that $x \\in h^{n} (U)$.\nChoose $r \\in C (X)_{+}$ such that $r (y) = 0$\nfor all $y \\in \\bigcup_{m = 0}^{n - 1} h^m (Y)$\nand $r (x) = 1$.\nThen, letting $u \\in C^* \\big( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\big)$\nbe the standard unitary (as in Notation~\\ref{ntn1-181221D}),\nwe have $(r \\otimes 1_D) u^n \\in B$\nby Lemma~\\ref{L_4X21_AYStruct}.\nTaking $l_x = r^2 \\cdot (f \\circ h^{- n})$,\nwe have $l_x (x) = f ( h^{- n} (x)) > \\frac{1}{2}$.\nAlso, $(r u^n) (f \\otimes 1_D) (r u^n)^* = l_x \\otimes 1_D$,\nso $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$.\n\nFinally, suppose $x \\not\\in U$\nand $x \\in \\bigcup_{m = 0}^{\\infty} h^m (Y)$.\nThen $x \\not\\in \\bigcup_{m = 1}^{\\infty} h^{- m} (Y)$.\nBy minimality of~$h$,\nthere is $n \\in {\\mathbb{Z}}_{> 0}$ such that $x \\in h^{- n} (U)$.\nChoose $r \\in C (X)_{+}$ such that $r (y) = 0$\nfor all $y \\in \\bigcup_{m = 1}^{n} h^{- m} (Y)$\nand $r (x) = 1$.\nThen $(r \\otimes 1_D) u^{- n} \\in B$\nby Lemma~\\ref{L_4X21_AYStruct}.\nTaking $l_x = r^2 \\cdot (f \\circ h^{n})$,\nwe have $l_x (x) = f ( h^{n} (x)) > \\frac{1}{2}$.\nAlso, $(r u^{- n}) (f \\otimes 1_D) (r u^{- n})^* = l_x \\otimes 1_D$,\nso $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$.\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{L_9Z01_KeepLarge}\nLet $X$ be a compact Hausdorff space,\nlet $G$ be a discrete group,\nand let $D$ be a C*-algebra.\nLet $(g, x) \\mapsto g x$ be an action of $G$ on~$X$,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over $(g, x) \\mapsto g x$\nin the sense of Definition~\\ref{D_4Y12_OverX}.\nThen for every\n$b \\in C^*_{\\mathrm{r}} \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$\nand every $\\varepsilon > 0$,\nthere is a finite set $T \\S G$ and a nonempty open set $V \\S X$\nsuch that,\nwhenever $x \\in V$\nand $f \\in C (X)$ satisfy $0 \\leq f \\leq 1$\nand $f (g x) = 1$ for all $g \\in T$,\nthen $\\| f b \\| > \\| b \\| - \\varepsilon$.\n\\end{lem}\n\nIf $D$ is not unital, then the product $f b$ is realized via\nthe inclusion of $C (X)$ in\n$M \\bigl( C^* \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr) \\bigr)$.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_9Z01_KeepLarge}]\nLet $(g, x) \\mapsto \\alpha_{g, x}$ be as in Definition~\\ref{D_4Y12_OverX}.\n\nFix a faithful representation $\\rho$ of $D$ on a Hilbert space~$H$.\nThen for every $x \\in X$ there is a representation\n$\\rho \\circ {\\mathrm{ev}}_x \\colon C (X, D) \\to L (H)$.\nLet $(v_x, \\pi_x)$ be the corresponding regular covariant\nrepresentation of $\\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$\non $l^2 (G, H)$,\nand let\n$\\sigma_x \\colon C^* \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr) \\to l^2 (G, H)$\nbe its integrated form.\nWe identify $l^2 (G, H)$ with $l^2 (G) \\otimes H$,\nand we write $\\delta_g \\in l^2 (G)$ for the standard basis vector\ncorresponding to $g \\in G$.\nFor later use, we recall that if $S_1, S_2 \\S G$ are finite sets,\n$d \\in C (X, D) [G]$\nis given as $d = \\sum_{g \\in S_1} d_g u_g$\nwith $d_g \\in C (X, D)$ for $g \\in S_1$,\nand $\\xi \\in l^2 (G) \\otimes H$ has the form\n$\\xi = \\sum_{h \\in S_2} \\delta_h \\otimes \\xi_h$\nwith $\\xi_h \\in H$ for $h \\in S_2$,\nthen\n\\begin{equation}\\label{Eq_9X01_IndRepFormula}\n\\sigma_x (d) \\xi\n = \\sum_{g \\in S_1} \\sum_{h \\in S_2}\n \\delta_{g h} \\otimes\n \\rho \\bigl( \\alpha_{h^{-1} g^{-1}, x} (d_g (g h x)) \\bigr) \\xi_h.\n\\end{equation}\n\nThe representation $\\bigoplus_{x \\in X} \\rho \\circ {\\mathrm{ev}}_x$\nis a faithful representation of $C (X, D)$,\nso $\\bigoplus_{x \\in X} \\sigma_x$\nis a faithful representation of\n$C^*_{\\mathrm{r}} \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$.\nTherefore there exists $x_0 \\in X$ such that\n$\\| \\sigma_{x_0} (b) \\| > \\| b \\| - \\frac{\\varepsilon}{5}$.\nChoose $c \\in C (X, D) [G]$ such that\n$\\| b - c \\| < \\frac{\\varepsilon}{5}$.\nIn particular,\n$\\| \\sigma_{x_0} (c) \\| > \\| b \\| - \\frac{2 \\varepsilon}{5}$.\nChoose $\\xi \\in l^2 (G, H)$ with finite support such that\n$\\| \\xi \\| = 1$ and\n$\\| \\sigma_{x_0} (c) \\xi \\| > \\| b \\| - \\frac{3 \\varepsilon}{5}$.\nWrite $c = \\sum_{g \\in S_1} c_g u_g$\nand $\\xi = \\sum_{h \\in S_2} \\delta_h \\otimes \\xi_h$\nwith $S_1, S_2 \\S G$ finite,\n$c_g \\in C (X, D)$ for $g \\in S_1$,\nand $\\xi_h \\in H$ for $h \\in S_2$.\nUse Lemma~\\ref{L_9Y30_MixCont} to choose\nan open set $V \\S X$ such that $x_0 \\in V$\nand such that for all $x \\in V$, $g \\in S_1$, and $h \\in S_2$, we have\n\\begin{equation}\\label{Eq_9Z01_Choice}\n\\bigl\\| \\alpha_{h^{-1} g^{-1}, x} (c_g (g h x))\n - \\alpha_{h^{-1} g^{-1}, x_0} (c_g (g h x_0) ) \\bigr\\|\n < \\frac{\\varepsilon}{5 {\\mathrm{card}} (S_1) {\\mathrm{card}} (S_2) + 1}.\n\\end{equation}\nSet\n\\[\nT = \\bigl\\{ g h \\colon {\\mbox{$g \\in S_1$ and $h \\in S_2$}} \\bigr\\}.\n\\]\nNow let $x \\in V$,\nand suppose $f \\in C (X)$ satisfies $0 \\leq f \\leq 1$\nand $f (g x) = 1$ for all $g \\in T$.\nThe condition on~$f$\nand the formula~(\\ref{Eq_9X01_IndRepFormula})\nimply that $\\sigma_x (f c) \\xi = \\sigma_x (c) \\xi$.\nApplying~(\\ref{Eq_9X01_IndRepFormula}) again,\nand using~(\\ref{Eq_9Z01_Choice}) at the second step,\nwe get\n\\begin{equation}\\label{Eq_9Z01_xx0}\n\\| \\sigma_x (c) \\xi - \\sigma_{x_0} (c) \\xi \\|\n \\leq \\sum_{g \\in S_1} \\sum_{h \\in S_2}\n \\bigl\\| \\alpha_{h^{-1} g^{-1}, x} (c_g (g h x))\n - \\alpha_{h^{-1} g^{-1}, x_0} (c_g (g h x_0) ) \\bigr\\|\n < \\frac{\\varepsilon}{5}.\n\\end{equation}\nTherefore, using $\\| f \\| \\leq 1$ at the second step\nand $\\| b - c \\| < \\frac{\\varepsilon}{5}$ and the second and fifth steps,\nas well as (\\ref{Eq_9Z01_xx0}) at the fourth step,\n\\begin{align*}\n\\| f b \\|\n& \\geq \\| \\sigma_x (f b) \\|\n > \\| \\sigma_x (f c) \\xi \\| - \\frac{\\varepsilon}{5}\n = \\| \\sigma_x (c) \\xi \\| - \\frac{\\varepsilon}{5}\n > \\| \\sigma_{x_0} (c) \\xi \\| - \\frac{2 \\varepsilon}{5}\n\\\\\n& > \\| \\sigma_{x_0} (b) \\xi \\| - \\frac{3 \\varepsilon}{5}\n > \\| \\sigma_{x_0} (b) \\| - \\frac{4 \\varepsilon}{5}\n > \\| \\sigma_{x_0} (b) \\| - \\frac{4 \\varepsilon}{5}\n > \\| b \\| - \\varepsilon,\n\\end{align*}\nas desired.\n\\end{proof}\n\n\\begin{prp}\\label{T_9921_PICase}\nLet $X$ be a compact metric space,\nlet $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $Y \\subset X$ be a compact subset\nsuch that $h^{n} (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\set{0}$,\nlet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nThen\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type.\n\\end{prp}\n\n\\begin{proof}\nWe verify directly the conditions of Definition~4.9 of~\\cite{PhLg}.\nWe follow Notation~\\ref{N_4Y12_CPN}.\nSet\n\\[\nA = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr),\n\\qquad\nB = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y,\n\\]\n\\[\nC = C (X, D),\n\\qquad {\\mbox{and}} \\qquad\nG = \\{ u \\}.\n\\]\nAll parts of condition~(1)\nin Definition~4.9 of~\\cite{PhLg}\nare obvious.\n\nWe verify condition~(2) there.\nLet $m \\in {\\mathbb{Z}}_{> 0}$,\nlet $a_1, a_2, \\ldots, a_m \\in A$,\nlet $\\varepsilon > 0$,\nlet $b_1 \\in A_{+}$ satisfy $\\| b_1 \\| = 1$,\nand let $b_2 \\in B_{+} \\setminus \\{ 0 ]$.\nWe will part of the verification of\ncondition~(3)\nin Proposition 4.11 of~\\cite{PhLg}\nin the proof of Theorem~\\ref{T_5418_AYLg}.\nChoose $c_1, c_2, \\ldots, c_m \\in C (X, D) [{\\mathbb{Z}}]$\nsuch that $\\| c_j - a_j \\| < \\varepsilon$\nfor $j = 1, 2, \\ldots, m$.\n(This estimate is condition~(2b).)\nChoose $N \\in {\\mathbb{Z}}_{> 0}$\nsuch that for $j = 1, 2, \\ldots, m$\nthere are $c_{j, l} \\in C (X, D)$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$\nwith\n\\[\nc_j = \\sum_{l = -N}^N c_{j, l} u^l.\n\\]\nApply Lemma~\\ref{L_9Z01_KeepLarge},\ngetting a finite set $T \\S {\\mathbb{Z}}$ and a nonempty open set $U \\S X$\nsuch that,\nwhenever $x \\in U$\nand $f \\in C (X)$ satisfy $0 \\leq f \\leq 1$\nand $f (h^k (x)) = 1$ for all $k \\in T$,\nthen $\\| f b_1^{1 \/ 2} \\| > 1 - \\frac{\\varepsilon}{2}$.\nDefine\n\\[\nS = \\bigl\\{ n - k \\colon\n {\\mbox{$k \\in T$ and $n \\in [- N, N] \\cap {\\mathbb{Z}}$}} \\bigr\\},\n\\]\nwhich is a finite subset of~${\\mathbb{Z}}$.\nThen $\\bigcup_{n \\in S} h^n (Y)$ is nowhere dense in~$X$,\nso there exists $x_0 \\in W$\nsuch that $x_0 \\not\\in \\bigcup_{n \\in S} h^n (Y)$.\nThis choice implies that\n\\[\n\\{ h^k (x_0) \\colon k \\in T \\} \\cap \\bigcup_{n = - N}^{N} h^n (Y)\n = \\varnothing,\n\\]\nso there is $g \\in C (X)$\nsuch that $0 \\leq g \\leq 1$,\n$g (y) = 1$ for all $y \\in \\bigcup_{n = - N}^{N} h^n (Y)$,\nand $g (h^k (x)) = 0$ for all $k \\in T$.\n\nCondition~(1a) in Definition~4.9 of~\\cite{PhLg}\nholds by construction.\nThe proof that $(1 - g) c_j \\in B$ for $j = 1, 2, \\ldots, m$\n(condition~(2c) in Definition~4.9 of~\\cite{PhLg})\nis the same as at the end of the proof of Theorem~\\ref{T_5418_AYLg}:\nfollow\nthe corresponding part of the proof of Theorem~7.10 of~\\cite{PhLg},\nexcept using \\Lem{L_4X21_AYStruct}\nin place of Proposition~7.5 of~\\cite{PhLg}.\nLemma~\\ref{L_0101_SubalgPI}\nimplies $1_A \\precsim_A b_1$ and $1_A \\precsim_B b_2$,\nso the requirements $g \\precsim_A b_1$ and $g \\precsim_B b_2$\n(condition~(2d) in Definition~4.9 of~\\cite{PhLg})\nfollow immediately.\nFor condition~(2e) in Definition~4.9 of~\\cite{PhLg},\nwe estimate, using $1 - g (h^k (x_0)) = 1$\nfor all $k \\in T$ at the second step:\n\\[\n\\bigl\\| (1 - g) b_1 (1 - g) \\bigr\\|\n = \\| (1 - g) b_1^{1 \/ 2} \\|^2\n > \\bigl( 1 - \\frac{\\varepsilon}{2} \\bigr)^2\n > \\varepsilon.\n\\]\nThis completes the proof.\n\\end{proof}\n\n\\section{Recursive structure for orbit breaking\n subalgebras}\\label{RecStruct}\n\nIn this section, under appropriate conditions\nwe will show that the orbit breaking subalgebras\nof Definition~\\ref{D_4Y12_OrbSubalg}\nhave a tractable recursive structure,\nas subalgebras of certain homogeneous algebras\ntensored with $D$. We start by introducing the\nformalism for a generalization of the recursive\nsubhomogeneous algebras introduced in\n\\cite{PhRsha1} that were crucial for the analysis in\n\\cite{QLinPhDiff} and \\cite{HLinPh}.\n\n\\begin{dfn}\\label{pullback}\nLet $A$, $B$, and $C$ be C*-algebra{s}, and let $\\varphi \\colon\nA \\to C$ and $\\psi \\colon B \\to C$ be homomorphisms.\nThen the associated {\\emph{pullback C*-algebra{}}} $A\n\\oplus_{C, \\varphi, \\psi} B$ is defined by\n\\[\nA \\oplus_{C, \\varphi, \\psi} B\n = \\bset{ (a, b) \\in A \\oplus B \\colon \\varphi (a) = \\psi (b) }.\n\\]\n\\end{dfn}\n\nWe frequently write $A \\oplus_{C} B$ when the maps $\\varphi$\nand $\\psi$ are understood.\nIf $C = 0$ we just get $A \\oplus B$.\n\n\\begin{dfn}\\label{RSHA}\nLet $D$ be a simple unital C*-algebra.\nThe class of\n{\\emph{recursive subhomogeneous algebras over $D$}} is\nthe smallest class $\\mathcal{R}$ of C*-algebra{s} that is\nclosed under isomorphism and such that:\n\\begin{enumerate}\n\\item\\label{9223_RSHA_1}\nIf $X$ is a compact Hausdorff space and $n \\geq 1$,\nthen $C (X, M_{n} (D)) \\in \\mathcal{R}$.\n\\item\\label{9223_RSHA_2}\nIf $B \\in \\mathcal{R}$, $X$ is compact Hausdorff,\n$n \\geq 1$, $X^{(0)} \\subset X$ is closed (possibly empty),\n$\\varphi \\colon B \\to C (X^{(0)}, \\, M_{n} (D))$\nis any unital homomorphism\n(the zero homomorphism{} if $X^{(0)}$ is empty), and\n$\\rho \\colon C (X, M_{n} (D)) \\to C (X^{(0)}, \\, M_{n} (D))$ is the\nrestriction homomorphism, then the pullback\n\\begin{align*}\n& B \\oplus_{C (X^{(0)}, \\, M_{n} (D)), \\, \\varphi, \\, \\rho)} C (X, M_{n} (D))\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n = \\bset{ (b, f) \\in B \\oplus C (X, M_{n} (D)) \\colon\n \\varphi (b) = f |_{(X^{(0)}} }\n\\end{align*}\nis in $\\mathcal{R}$.\n\\end{enumerate}\n\\end{dfn}\n\nTaking $D = {\\mathbb{C}}$ in this definition gives the usual definition\nfor the class of recursive subhomogeneous algebras.\n(See \\cite{PhRsha1}.)\nThis definition makes sense for any unital C*-algebra~$D$,\nbut it is not clear whether it is appropriate in this generality.\n\n\\begin{dfn}\\label{RSHAMachinery}\nWe adopt the following standard notation for recursive\nsubhomogeneous algebras over $D$.\nThe definition implies\nthat if $R$ is any recursive subhomogeneous algebra over\n$D$, then $R$ may be written in the form\n\\[\nR \\cong\n \\left[ \\cdots \\left[ \\left[ C_{0} \\oplus_{C_{1}^{(0)}, \\varphi_1, \\rho_1}\n C_{1} \\right]\n \\oplus_{C_{2}^{(0)}, \\varphi_2, \\rho_2} \\right] \\cdots \\right]\n \\oplus_{C_{l}^{(0)}, \\varphi_l, \\rho_l} C_{l},\n\\]\nwith $C_{k} = C (X_{k}, \\, M_{n (k)} (D))$ for compact Hausdorff\nspaces $X_{k}$ and positive integers $n (k)$, and with\n$C_{k}^{(0)} = C \\bigl( X_{k}^{(0)}, \\, M_{n (k)} (D) \\bigr)$\nfor compact\nsubsets $X_{k}^{(0)} \\subset X_{k}$ (possibly empty),\nwhere the maps $\\rho_{k} \\colon C_{k} \\to C_{k}^{(0)}$ are\nalways the restriction maps.\nAn expression of this type for\n$R$ will be referred to as a {\\emph{decomposition}} of~$R$,\nand the notation that appears here will be referred to as the\n{\\emph{standard notation for a decomposition}}.\nWe associate the following objects to this decomposition.\n\\begin{enumerate}\n\\item\\label{9223_RSHAMachinery_1}\nIts {\\emph{length}} $l$.\n\\item\\label{9223_RSHAMachinery_2}\nThe {\\emph{$k$-th stage algebra}}\n\\[\nR^{(k)}\n = \\left[ \\cdots \\left[ \\left[ C_{0} \\oplus_{C_{1}^{(0)}}\n C_{1} \\right] \\oplus_{C_{2}^{(0)}} C_{2} \\right] \\cdots \\right]\n \\oplus_{C_{k}^{(0)}} C_{k}.\n\\]\n\\item\\label{9223_RSHAMachinery_3}\nIts {\\emph{base spaces}} $X_{0},\\ldots,X_{l}$ and {\\emph{total space}}\n$X = \\coprod_{k = 0}^{l} X_{k}$.\n\\item\\label{9223_RSHAMachinery_4}\nIts {\\emph{matrix sizes}} $n (0),\\ldots,n (l)$ and {\\emph{matrix\nsize function}} $m \\colon X \\to {\\mathbb{Z}}_{\\geq 0}$ defined by $m(x) = n (k)$\nwhen $x \\in X_{k}$.\n(This is called the {\\emph{matrix size of $R$ at~$x$}}.)\n\\item\\label{9223_RSHAMachinery_5}\nIts {\\emph{minimum matrix size}} $\\min_{k} n (k)$ and\n{\\emph{maximum matrix size}} $\\max_{k} n (k)$.\n\\item\\label{9223_RSHAMachinery_6}\nIts {\\emph{topological dimension}}\n$\\dim (X) = \\max_{k} \\dim (X_{k})$\nand {\\emph{topological dimension function}}\n$d \\colon X \\to {\\mathbb{Z}}_{\\geq 0}$, defined by $d(x) = \\dim (X_{k})$\nfor $x \\in X_{k}$.\n(This is called the {\\emph{topological dimension of $R$ at $x$}}.)\n\\item\\label{9223_RSHAMachinery_7}\nIts {\\emph{standard representation}}\n$\\sigma\n = \\sigma_{R} \\colon R\n \\to \\bigoplus_{k = 0}^{l} C (X_{k}, \\, M_{n (k)} (D))$,\ndefined by\nforgetting the restriction to a subalgebra in each of the fibered\nproducts in the decomposition.\n\\item\\label{9223_RSHAMachinery_8}\nThe associated {\\emph{evaluation maps}}\n${\\mathrm{ev}}_{x} \\colon R \\to M_{n (k)} (D)$,\ndefined to be the restriction of the usual evaluation\nmap on $\\bigoplus_{k = 0}^{l} C (X_{k}, \\, M_{n (k)} (D))$ to $R$,\nwhen $R$\nis identified with a subalgebra of this algebra through the standard\nrepresentation $\\sigma_{R}$.\n\\end{enumerate}\n\\end{dfn}\n\nA decomposition of an algebra $R$ as a\nrecursive subhomogeneous algebra over $D$ may be highly\nnonunique, which in the case $D = {\\mathbb{C}}$ is clear from examples\nin~\\cite{PhRsha1}.\nMoreover, the matrix sizes are not uniquely\ndetermined even if all the other data has already been chosen,\nwhich is easily seen through\nthe example of the $2^{\\infty}$~UHF algebra.\n\n\\begin{ntn}\\label{StronglySelfAbs}\nThroughout, we let $\\mathcal{E}$ denote a separable, unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}:\nthere\nexists an isomorphism $\\mathcal{E} \\cong \\mathcal{E} \\otimes \\mathcal{E}$ that is unitarily\nequivalent to $a \\mapsto a \\otimes 1_{\\mathcal{E}}$.\n\\end{ntn}\n\nExamples include the Cuntz algebras\n$\\mathcal{O}_{2}$ and $\\mathcal{O}_{\\infty}$,\nand the Jiang-Su algebra ${\\mathcal{Z}}$,\nwhich is a simple, separable, unital, infinite dimensional,\nstrongly selfabsorbing, nuclear C*-algebra{} having the same\nElliott invariant\nas the complex numbers ${\\mathbb{C}}$.\n(See \\cite{JiangSu} for its construction).\n\n\\begin{dfn}\\label{DStableDefn}\nAdopt Notation~\\ref{StronglySelfAbs}.\nA separable C*-algebra{} $D$ is called {\\emph{$\\mathcal{E}$-stable}}\nif there is\nan isomorphism $D \\otimes \\mathcal{E} \\cong D$.\n\\end{dfn}\n\nIt is clear that if $D$ is $\\mathcal{E}$-stable, then so are $M_{n} (D)$\nand $C (X, D)$.\nWith appropriate assumptions on the algebra $D$, we can give some\nresults about the $\\mathcal{E}$-stability of recursive subhomogeneous algebras\nover $C$.\n\n\\begin{lem}\\label{DStablePullback}\nAdopt Notation~\\ref{StronglySelfAbs}.\nLet $A$, $B$, and $C$ be separable unital C*-algebra{s}\n(allowing $C = 0$),\nand let $\\varphi \\colon A \\to C$ and $\\psi \\colon B \\to C$\nbe unital homomorphisms.\nLet $P = A \\oplus_{C, \\varphi, \\psi} B$ be the associated pullback\n(Definition~\\ref{pullback}).\nIf $\\psi$\nis surjective and both $A$ and $B$ are $\\mathcal{E}$-stable, then $P$ is\n$\\mathcal{E}$-stable.\n\\end{lem}\n\n\\begin{proof}\nDefine\n$\\pi \\colon P \\to A$ by $\\pi (a, b) = a$.\nThere is an isomorphism\n$\\iota \\colon \\mathrm{Ker} (\\psi) \\to \\mathrm{Ker} (\\pi)$\ngiven by $\\iota (b) = (0, b)$ for $b \\in \\mathrm{Ker} (\\psi)$.\nMoreover, surjectivity of $\\psi$ is easily seen to imply\nsurjectivity of $\\pi$.\nWe obtain an exact\nsequence\n\\[\n0 \\longrightarrow \\mathrm{Ker} (\\psi)\n \\stackrel{\\iota}{\\longrightarrow} P\n \\stackrel{\\pi}{\\longrightarrow} A\n \\longrightarrow 0.\n\\]\nNow $\\mathrm{Ker} (\\varphi)$ is $\\mathcal{E}$-stable by Corollary 3.1 of\n\\cite{TomsWinter2}.\nSince $B$ is also $\\mathcal{E}$-stable, Theorem 4.3 of\n\\cite{TomsWinter2} implies that $P$ is $\\mathcal{E}$-stable.\n\\end{proof}\n\n\n\\begin{prp}\\label{DStableRSHA}\nAdopt Notation~\\ref{StronglySelfAbs}.\nLet $D$ be a simple separable $\\mathcal{E}$-stable C*-algebra,\nand let $R$ be a recursive subhomogeneous algebra over $D$.\nThen $R$ is $\\mathcal{E}$-stable.\n\\end{prp}\n\n\\begin{proof}\nWe proceed by induction on the length of a decomposition for $R$ as\na recursive subhomogeneous algebra over $D$.\nThe base case is when\n$R = C (X, M_{n} (D))$, and this is $\\mathcal{E}$-stable whenever $D$ is\n$\\mathcal{E}$-stable.\nFor the inductive step, we may assume that\nthere are an $\\mathcal{E}$-stable unital C*-algebra~$R'$,\n$n \\in {\\mathbb{Z}}_{> 0}$, a compact Hausdorff space~$X$,\na closed subset $X^{(0)} \\subset X$,\nand a unital homomorphism\n$\\varphi \\colon R' \\to C \\bigl( X^{(0)}, \\, M_{n} (D) \\bigr)$\nsuch that\n\\[\nR = \\bset{ (a, f) \\in R' \\oplus C (X, M_{n} (D))\n \\colon \\varphi (a) = f |_{X^{(0)}} }\n\\]\nSince $f \\mapsto f |_{X^{(0)}}$ is surjective and both\n$R'$ and $C (X, M_{n} (D))$\nare $\\mathcal{E}$-stable, it follows from Lemma~\\ref{DStablePullback}\nthat $R$ is $\\mathcal{E}$-stable,\nwhich completes the induction.\n\\end{proof}\n\n\\begin{prp}\\label{AyDirLim}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed, and let\n$Y_{1} \\supset Y_{2} \\supset \\cdots$\nbe closed subsets of $X$ such that $\\bigcap_{n = 1}^{\\infty}\nY_{n} = Y$.\nThen\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n = {\\overline{ {\\ts{ {\\ds{\\bigcup}}_{n = 1}^{\\infty} }} A_{Y_{n}} }}\n \\cong \\varinjlim A_{Y_{n}}.\n\\]\n\\end{prp}\n\n\\begin{proof}\nThe proof is easy, and is omitted.\n\\end{proof}\n\nThe results in the remainder of this section are\nmostly generalizations of ones in Section 1 of\n\\cite{QLinPhDiff}.\nWe follow a slightly more modern development,\nadapted from Section 11.3 of \\cite{PhNotes}, since \\cite{QLinPhDiff}\nhas not been published.\n(The proofs in \\cite{PhNotes} are mostly\ntaken from \\cite{QLinPhDiff}, with some changes in notational\nconventions.)\nMost proofs go through with changes only to the notation,\nsuch as replacing the action of a minimal homeomorphism\n$h$ with the action $\\alpha$ on $C (X, D)$.\nThe biggest technical\ndifferences are in the proof of Lemma~\\ref{L_4X21_AYStruct}.\nWe begin with the well-known Rokhlin tower construction.\n\n\\begin{ntn}\\label{ModifiedRokhlinTower}\nLet $X$ be a compact Hausdorff space{} and let $h \\colon X \\to X$ be a minimal homeomorphism.\nLet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nFor $y \\in Y$, define\n$r (y) = \\inf \\bigl( \\set{m \\geq 1 \\colon h^{m} (y) \\in Y} \\bigr)$.\nThen $\\sup_{y \\in Y} r (y) < \\infty$\n(see Lemma \\ref{RokhlinTowerProps}(\\ref{9Z26_new})),\nso there are finitely many\ndistinct values $n (0) < n (1) < \\cdots < n (l)$ in the range of~$r$.\nFor $k = 0, 1, \\ldots l$, set\n\\[\nY_{k} = \\overline{ \\set{ y \\in Y \\colon r (y) = n (k) } }\n\\qquad {\\mbox{and}} \\qquad\nY_{k}^{\\circ}\n = {\\mathrm{int}} \\bigl( \\bset{ y \\in Y \\colon r (y) = n (k) } \\bigr).\n\\]\n\\end{ntn}\n\nWe warn that, while\n$Y_{k}^{\\circ} \\subset {\\mathrm{int}} (Y_k)$,\nthe inclusion may be proper.\n\n\\begin{lem}\\label{RokhlinTowerProps}\nLet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nThen:\n\\begin{enumerate}\n\\item\\label{9Z26_new}\n$\\sup_{y \\in Y} r (y) < \\infty$.\n\\item\\label{Item_RokhlinTowerProps_1}\nThe sets $h^{j} (Y_{k}^{\\circ})$ are pairwise disjoint for $0\n\\leq k \\leq l$ and $1 \\leq j \\leq n (k)$.\n\\item\\label{Item_RokhlinTowerProps_2}\n$\\bigcup_{k = 0}^{l} Y_{k} = Y$.\n\\item\\label{Item_RokhlinTowerProps_3}\n$\\bigcup_{k = 0}^{l} \\bigcup_{j= 0}^{n (k)-1} h^{j} (Y_{k}) = X$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nThis is Lemma 11.3.5 of~\\cite{PhNotes}.\n\\end{proof}\n\nWe first consider the specific situation of the structure of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nwhen $X$ is the Cantor set.\nThis is needed later.\n\n\\begin{lem}\\label{CantorSetCase}\nAdopt Notation \\ref{ModifiedRokhlinTower}, let $X$ be the Cantor\nset, and let $Y \\subset X$ be a nonempty compact open subset.\nThen\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\cong \\bigoplus_{k = 0}^{l} C (Y_{k}, \\, M_{n (k)} (D)).\n\\]\n\\end{lem}\n\n\\begin{proof}\nThis is a straightforward adaption of the proof of Lemma 11.2.20\nof~\\cite{PhNotes}.\n\\end{proof}\n\nIf we set $B_{k} = C (Y_{k}, M_{n (k)})$, then $B_{k}$ is an\nAF algebra for each~$k$\n(since $Y_{k}$ is totally disconnected), and hence\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a direct sum of what might be called ``AF $D$-algebras''.\n\n\\begin{prp}\\label{BanachSpaceDirSum}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nlet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nFollowing the notation of Lemma~\\ref{L_4X21_AYStruct}, there is\n$N \\in {\\mathbb{Z}}_{> 0}$ such that $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nhas the Banach space direct sum decomposition\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n = \\bigoplus_{n = - N}^{N} C_{0} (X \\setminus Y_{n}, \\, D) u^{n}.\n\\]\n\\end{prp}\n\n\\begin{proof}\nThe proof is the same as that of Corollary 11.3.7 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{prp}\\label{AyHom}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nlet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nDefine the unitary $s_{k} \\in C (Y_{k}, M_{n (k)} (D))$ by\n\\[\ns_{k} = \\left[ \\begin{array}{ccccccc}\n0 & 0 & \\cdots & \\cdots & 0 & 0 & 1 \\\\\n1 & 0 & \\cdots & \\cdots & 0 & 0 & 0 \\\\\n0 & 1 & \\cdots & \\cdots & 0 & 0 & 0 \\\\\n\\vdots & \\vdots & \\ddots & & \\vdots & \\vdots & \\vdots \\\\\n\\vdots & \\vdots & & \\ddots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & \\cdots & 1 & 0 & 0 \\\\\n0 & 0 & \\cdots & \\cdots & 0 & 1 & 0 \\end{array} \\right].\n\\]\nFor $k = 0, 1, \\ldots, l$ there is a unique linear map\n\\[\n\\gamma_{k} \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\to C (Y_{k}, M_{n (k)} (D))\n\\]\nsuch that for $m = 0, 1, \\ldots, n (k) - 1$\nand $f \\in C_{0} (X \\setminus Z_{m}, D)$ we have:\n\\begin{enumerate}\n\\item\\label{9222_AyHom_1}\n$\\gamma_{k} (f u^{m}) = {\\mathrm{diag}} \\bigl( f |_{Y_{k}}, \\,\n \\alpha^{-1} (f) |_{Y_{k}}, \\,\n \\ldots, \\, \\alpha^{-[n (k)-1]} (f) |_{Y_{k}} \\bigr) \\cdot s_{k}^{m}$.\n\\item\\label{9222_AyHom_2}\n$\\gamma_{k} (u^{-m} f) = s_{k}^{-m} \\cdot\n {\\mathrm{diag}} \\bigl( f |_{Y_{k}}, \\, \\alpha^{-1} (f)\n |_{Y_{k}}, \\, \\ldots, \\,\n \\alpha^{-[n (k)-1]} (f) |_{Y_{k}} \\bigr)$.\n\\end{enumerate}\nMoreover, the map\n\\[\n\\gamma \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\to \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))\n\\]\ngiven by $\\gamma (a) = (\\gamma_{0} (a), \\gamma_{1} (a), \\ldots, \\gamma_{l} (a))$\nis a\n$*$-homomorphism.\n\\end{prp}\n\n\\begin{proof}\nThe computations in this proof are analogous to those in\nthe proof of Proposition 11.3.9 of~\\cite{PhNotes}, using\nLemma~\\ref{L_4X21_AYStruct}\nin place of Proposition 11.3.6 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{lem}\\label{SubdiagProjnDecomp}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nand let $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nAdopt Notation \\ref{ModifiedRokhlinTower}.\nIdentify $C (Y_{k}, \\, M_{n (k)} (D))$\nwith $M_{n (k)} (C (Y_{k}, D))$ in the\nobvious way.\nDefine maps\n\\[\nE_{k}^{(m)} \\colon C (Y_{k}, M_{n (k)} (D))\n\\to C (Y_{k}, M_{n (k)} (D))\n\\]\nby $E_{k}^{(m)} (b)_{m+j,j} = b_{m+j,j}$\nfor $1 \\leq j \\leq n (k) - m$ (if $m \\geq 0$)\nand for $-m + 1 \\leq j \\leq n (k)$ (if $m \\leq 0$),\nand $E_{k}^{(m)} (b)_{i,j} = 0$ for\nall other pairs $(i,j)$.\n(Thus, $E_{k}^{(m)}$ is the projection map\non the $m$-th subdiagonal.)\nWrite\n\\[\nG_{m} = \\bigoplus_{k = 0}^{l} E_{k}^{(m)} (C (Y_{k}, M_{n (k)} (D)).\n\\]\nThen:\n\\begin{enumerate}\n\\item\\label{SubdiagProjnDecomp_DirSum}\nThere is a Banach space direct sum decomposition\n\\[\n\\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D)) =\n\\bigoplus_{m=-n (l)}^{n (l)} G_{m}.\n\\]\n\\item\\label{SubdiagProjnDecomp_gmIs}\nFor $k = 0, 1,\\ldots,l$, $m \\geq 0$,\n$f \\in C_{0} (X \\setminus Z_{m}, D)$,\nand $x \\in Y_{k}$,\nthe expression $\\gamma_{k} (f u^{m}) (x)$\nis given by the following matrix, in which the first nonzero\nentry is in row $m+1$:\n\\[\n\\gamma_{k} (f u^{m}) (x)\n = \\left[ \\begin{array}{ccccccc} 0 & 0 & \\cdots &\n\\cdots & \\cdots & \\cdots & 0\n\\\\\n\\vdots & \\vdots & & & & & \\vdots \\\\\n0 & 0 & & & & & \\vdots\n\\\\\n\\alpha^{-m} (f) (x) & 0 & & & & & \\vdots \\\\\n0 & \\alpha^{-[m+1]} (f) (x) & & & & & \\vdots\n\\\\\n\\vdots & & \\ddots & & &\n& \\vdots\n\\\\\n0 & \\cdots & \\cdots & \\alpha^{-[n (k)-1]} (f) (x) & 0\n& \\cdots & 0\n\\end{array} \\right].\n\\]\n\\item\\label{SubdiagProjnDecomp_3}\nFor $m \\geq 0$ and $f \\in C_{0} (X \\setminus Z_{m}, D)$, we\nhave\n\\[\n\\gamma_{k} (f u^{m}) \\in E_{k}^{(m)} (C (Y_{k}, M_{n (k)} (D))),\n\\qquad\n\\gamma (f u^{m}) \\in G_{m},\n\\]\n\\[\n\\gamma_{k} (u^{-m} f) \\in E_{k}^{(-m)} (C (Y_{k}, M_{n (k)} (D))),\n\\qquad {\\mbox{and}} \\qquad\n\\gamma (u^{-m} f) \\in G_{-m}.\n\\]\n\\item\\label{SubdiagProjnDecomp_Compatible}\nThe homomorphism $\\gamma$ is compatible with the\ndirect sum decomposition of Proposition~\\ref{BanachSpaceDirSum}\non its domain and the direct sum decomposition of part (1) on its\ncodomain.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nThe direct sum decomposition is essentially immediate from the\ndefinition of the maps $E_{k}^{(m)}$, while the other statements\nfollow from Proposition~\\ref{BanachSpaceDirSum} and some\nstraightforward matrix calculations as in the proof of Lemma 11.3.15\nof~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{cor}\\label{GammaInj}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen\nthe homomorphism $\\gamma$ of Proposition~\\ref{BanachSpaceDirSum}\nis injective.\n\\end{cor}\n\n\\begin{proof}\nThe proof is the same as the proof of Lemma 11.3.17 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{lem}\\label{ByMemberCond}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nLet\n\\[\nb = (b_{0}, b_{1}, \\ldots, b_{l})\n \\in \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D)).\n\\]\nThen\n$b \\in \\gamma \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr)$\nif and only if whenever\n\\begin{itemize}\n\\item\n$r > 0$,\n\\item\n$k,t_{1},\\ldots,t_{r} \\in \\set{0,\\ldots,l}$,\n\\item\n$n (t_{1}) + n (t_{2}) + \\cdots + n (t_{r}) = n (k)$,\n\\item\n$x \\in (Y_{k} \\setminus Y_{k}^{\\circ}) \\cap\nY_{t_{1}} \\cap h^{-n (t_{1})} (Y_{t_{2}}) \\cap \\cdots \\cap\nh^{-[n (t_{1}) + \\cdots + n (t_{r - 1}) ]} (Y_{t_{r}})$,\n\\end{itemize}\nthen $b_{k} (x)$ is given by the block\ndiagonal matrix\n\\[\nb_{k} (x) = \\left[ \\begin{array}{cccc}\nb_{t_{1}} (x) & & &\n\\\\\n& \\alpha^{-n (t_{1})} (b_{t_{2}}) (x) & &\n\\\\\n& &\n\\ddots &\n\\\\\n& & & \\alpha^{-[n (t_{1}) + \\cdots + n (t_{r - 1})]}\n(b_{t_{r}}) (x) \\end{array} \\right].\n\\]\n\\end{lem}\n\n\\begin{proof}\nThe proof is analogous (with appropriate changes to notation\nand exponents) to the proof of Lemma 11.3.18 of~\\cite{PhNotes}.\n\\end{proof}\n\nWe are now in position to give a decomposition of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nas a recursive subhomogeneous algebra over $D$.\n\n\\begin{thm}\\label{AyRSHA}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nAdopt Notation \\ref{ModifiedRokhlinTower} and the notation of\nProposition \\ref{AyHom}.\nThen the homomorphism\n\\[\n\\gamma \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\to \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))\n\\]\nof Proposition \\ref{AyHom}\ninduces an isomorphism of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nwith the recursive subhomogeneous algebra over~$D$ defined,\nin the notation of Definition \\ref{RSHAMachinery}, as follows:\n\\begin{enumerate}\n\\item\\label{9223_AyRSHA_1}\n$l$ and $n (0),n (1),\\ldots,n (l)$ are as in Notation\n\\ref{ModifiedRokhlinTower}.\n\\item\\label{9223_AyRSHA_2}\n$X_{k} = Y_{k}$ for $0 \\leq k \\leq l$.\n\\item\\label{9223_AyRSHA_3}\n$X_{k}^{(0)} = Y_{k} \\cap \\bigcup_{j= 0}^{k - 1} Y_{j}$.\n\\item\\label{9223_AyRSHA_4}\nFor $x \\in X_{k}^{(0)}$ and $b = (b_{0}, b_{1},\\ldots, b_{k - 1})$\nin the image in $\\bigoplus_{j= 0}^{k - 1} C (Y_{j}, M_{n (j)} (D))$\nof the $k - 1$ stage algebra\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y^{(k - 1)}$, whenever\n\\[\nx \\in (Y_{k} \\setminus Y_{k}^{\\circ}) \\cap Y_{t_{1}} \\cap\nh^{-n (t_{1})} (Y_{t_{2}}) \\cap \\cdots \\cap h^{-[n (t_{1}) + \\cdots +\nn (t_{r - 1}]} (Y_{t_{r}})\n\\]\nwith\n$n (t_{1}) + n (t_{2}) + \\cdots + n (t_{r}) = n (k)$,\nthen $\\varphi_{k} (b (x))$ is given by the block\ndiagonal matrix\n\\[\n\\varphi_{k} (b (x))\n = \\left[ \\begin{array}{cccc}\nb_{t_{1}} (x) & & &\n\\\\\n& \\alpha^{-n (t_{1})} (b_{t_{2}}) (x) & &\n\\\\\n& & \\ddots &\n\\\\\n& & & \\alpha^{-[n (t_{1}) + \\cdots + n (t_{r - 1})]}\n(b_{t_{r}}) (x) \\end{array} \\right].\n\\]\n\\item\\label{9223_AyRSHA_5}\n$\\rho_{k}$ is the restriction map.\n\\end{enumerate}\nThe topological dimension of this decomposition is $\\dim (X)$, and\nthe standard representation of\n$\\sigma \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr)$\nis the inclusion map in\n$\\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))$.\n\\end{thm}\n\n\\begin{proof}\nThe proof is analogous to the proof of Theorem 11.3.19\nof~\\cite{PhNotes}, while the proof of the statement regarding the\ntopological dimension is the same as for the proof given\nin~\\cite{PhNotes} for the part of Theorem 11.3.14 that is not\nincluded in Theorem 11.3.19.\n\\end{proof}\n\n\\section{Structural properties of the orbit breaking subalgebra\n and the crossed product}\\label{StructAY}\n\nIn this section,\nwe give some general theorems on the structure of C*-algebra{s}\nof the form $C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)$,\nwhich we derive from results on the structure of\norbit breaking subalgebras.\nWe give many explicit examples in Section~\\ref{Sec_InfRed}.\n\n\\begin{cnv}\\label{N_9215_AAY_ntn}\nIn this section,\nas in Definition~\\ref{D_4Y12_OrbSubalg}\nbut with additional restrictions,\n$X$ will be an infinite compact metric space,\n$h \\colon X \\to X$ will be a minimal homeomorphism,\n$D$ will be a simple unital C*-algebra,\nand $\\alpha \\in {\\mathrm{Aut}} ( C (X, D))$\nwill be an automorphism which lies over~$h$.\n\\end{cnv}\n\nFor a few results, simplicity is not needed.\n\n\\begin{prp}\\label{AyRSHADirLim}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nFor any nonempty closed set $Y \\subset X$,\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is a direct limit of\nrecursive subhomogeneous algebras over~$D$\nof the form $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_{Y_n}$\nfor closed subsets $Y_n \\S X$ with ${\\mathrm{int}} (Y_n) \\neq \\varnothing$.\n\\end{prp}\n\n\\begin{proof}\nGiven $Y \\subset X$ closed, choose a sequence\n$(Y_{n})_{n = 1}^{\\infty}$ of closed subsets of $X$\nwith ${\\mathrm{int}} (Y_{n}) \\neq \\varnothing$ and\n$Y_{n + 1} \\subset Y_{n}$ for $n \\in {\\mathbb{Z}}_{> 0}$,\nand with $\\bigcap_{n = 1}^{\\infty} Y_{n} = Y$.\nThat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is\na direct limit of separable recursive subhomogeneous algebras\nover $D$ follows by applying Theorem~\\ref{AyRSHA} and\nProposition~\\ref{AyDirLim}.\n\\end{proof}\n\n\\begin{prp}\\label{AYDStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $\\mathcal{E}$ be a separable unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}.\nAssume that $D$ is separable and $\\mathcal{E}$-stable.\nLet $Y \\subset X$ be closed and nonempty.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is $\\mathcal{E}$-stable.\n\\end{prp}\n\n\\begin{proof}\nIf ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nthis follows immediately from Theorem \\ref{AyRSHA} and\nProposition \\ref{DStableRSHA}.\nThe case of a general nonempty closed set\nnow follows from Proposition~\\ref{AyRSHADirLim} above\nand Corollary 3.4 of~\\cite{TomsWinter2}.\n\\end{proof}\n\nIn particular, we get ${\\mathcal{Z}}$-stability.\n\n\\begin{cor}\\label{EStableGivesZStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $\\mathcal{E}$ be a separable unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}.\nAssume that $D$ is separable and $\\mathcal{E}$-stable.\nLet $Y \\subset X$ be closed and nonempty.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is ${\\mathcal{Z}}$-stable.\n\\end{cor}\n\n\\begin{proof}\nStrongly selfabsorbing C*-algebra{s} are ${\\mathcal{Z}}$-stable\nby Theorem~3.1 of~\\cite{WinterZStab},\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is $\\mathcal{E}$-stable by\nProposition~\\ref{AYDStable}, so the conclusion is immediate.\n\\end{proof}\n\nWe now turn to ${\\mathcal{Z}}$-stability\nof the crossed product $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\nWhen $D$ is nuclear, it can be obtained from the results of\n\\cite{ABP-Zstab}.\nHowever, if nuclearity is not assumed, then the\nmain conclusion of that paper only implies\nthat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is tracially\n${\\mathcal{Z}}$-stable (in the sense of~\\cite{HirOr}).\n\n\\begin{thm}\\label{ATraciallyZStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nAssume that $D$ is a simple separable unital ${\\mathcal{Z}}$-stable\nC*-algebra{} which has a quasitrace.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis tracially ${\\mathcal{Z}}$-stable.\nIf, in addition, $D$ is nuclear,\nthen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is ${\\mathcal{Z}}$-stable.\n\\end{thm}\n\n\\begin{proof}\nTheorem 3.3 of~\\cite{HirOr} implies that $D$ has strict comparison\nof positive elements, and so\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ by\nCorollary \\ref{AyCentLargeConds}(\\ref{9212_AyCentLargeConds_StrCmp}).\nAlso, $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis ${\\mathcal{Z}}$-stable by Corollary~\\ref{EStableGivesZStable},\nso Theorem~2.2 of~\\cite{ABP-Zstab} implies\nthat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\ntracially ${\\mathcal{Z}}$-stable.\nIf $D$ is nuclear, then it is ${\\mathcal{Z}}$-stable\nby Theorem 4.1 of~\\cite{HirOr},\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is nuclear.\nSo ${\\mathcal{Z}}$-stability of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nfollows from Theorem 2.3\nof~\\cite{ABP-Zstab}.\n\\end{proof}\n\n\\begin{cor}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $D$ be a simple separable unital nuclear ${\\mathcal{Z}}$-stable\nC*-algebra{} which has a quasitrace.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ has nuclear\ndimension at most 1.\n\\end{cor}\n\n\\begin{proof}\nThis follows immediately from Theorem~\\ref{ATraciallyZStable} above\nand Theorem~B of~\\cite{CETWW}.\n\\end{proof}\n\nCorollary~\\ref{EStableGivesZStable} and its consequences\nrequire no\nassumption about the underlying dynamical system other than minimality.\nIn particular, if $D$ is ${\\mathcal{Z}}$-stable then\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is\n${\\mathcal{Z}}$-stable when $X$ is infinite dimensional, and even when\n$\\mathrm{mdim} (X, h)$ (as in Notation~\\ref{N_0108_DimX}) is strictly positive.\nIf moreover $D$ is nuclear\nthen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\n${\\mathcal{Z}}$-stable.\nThus, crossed products of the form\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ are\n${\\mathcal{Z}}$-stable\nwhenever $h$ is minimal and $D$ is\nsimple, separable, unital, nuclear, and ${\\mathcal{Z}}$-stable,\nregardless of anything else about the underlying dynamics.\n\nIf $D$ is not assumed to be ${\\mathcal{Z}}$-stable,\nthen we must make assumptions about the dynamical system\n$(X, h)$ besides just minimality to expect\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ to have\ntractable structure.\nThe hypotheses in Proposition~\\ref{TsrRR0A_Y},\nTheorem~\\ref{P_9Z26_XisCantor},\nProposition~\\ref{P_9Z21_dim1},\nand Theorem~\\ref{T_9222_Tsr1AndRR0_CrPrd}\n(which also include conditions on~$D$)\nare surely much stronger than needed.\nThey have the advantage that the proofs can easily by obtained from\nresults already in the literature.\n\n\\begin{prp}\\label{TsrRR0A_Y}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $X$ is the Cantor set.\nLet $Y \\subset X$ be closed and nonempty.\n\\begin{enumerate}\n\\item\\label{TsrRR0A_Y_tsr}\nIf ${\\mathrm{tsr}} (D) = 1$, then\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 1$.\n\\item\\label{TsrRR0A_Y_RR}\nIf ${\\mathrm{RR}} (D) = 0$, then\n${\\mathrm{RR}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 0$.\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nFirst suppose that ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen, from Lemma~\\ref{CantorSetCase} and the remark immediately\nafter it,\nwe see there are are AF algebras $B_{0}, B_{1}, \\ldots, B_{k}$\nsuch that\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\cong \\bigoplus_{k = 0}^{l} B_{k} \\otimes D.\n\\]\nThe result follows immediately.\n\nSince stable rank one and real rank zero are preserved by direct limits,\nthe general case now follows from\nProposition~\\ref{AyRSHADirLim}.\n\\end{proof}\n\n\\begin{thm}\\label{P_9Z26_XisCantor}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $X$ is the Cantor set.\nFurther assume that one of the following conditions holds:\n\\begin{enumerate}\n\\item\\label{9217_GetTsr1_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9217_GetTsr1_StrCmp}\n$D$ has strict comparison of positive elements.\n\\item\\label{9217_GetTsr1_SP}\n$D$ has property (SP) and the order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9217_GetTsr1_CptGen}\nThe set\n$\\bigl\\{ \\alpha_x^n \\colon {\\mbox{$x \\in X$ and $n \\in {\\mathbb{Z}}$}} \\bigr\\}$\nis contained in a subset of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nIf ${\\mathrm{tsr}} (D) = 1$, then\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\bigr) = 1$,\nand if also ${\\mathrm{RR}} (D) = 0$, then\n${\\mathrm{RR}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\bigr) = 0$.\n\\end{thm}\n\nThe action in this theorem has the Rokhlin property.\nHowever, it seems to be unknown whether the crossed product of\neven a\nsimple unital C*-algebra{} with stable rank one by a Rokhlin automorphism\nstill has stable rank one.\nSee Problem~\\ref{Pb_9922_RokhlinTsr} and the discussion afterwards.\n\nWe also emphasize that the conclusion for real rank zero\nis only valid if stable rank one is also assumed.\nThe reason is the stable rank one hypothesis\nin Theorem~6.4 of~\\cite{ArPh}.\nThat theorem likely holds without stable rank one,\nbut this has not yet been proved.\nFor now, this is no great loss,\nsince, other than purely infinite simple C*-algebra{s}\n(which are covered by our Theorem~\\ref{T_0101_PICase}),\nthere are no known examples of simple unital C*-algebra{s} which have\nreal rank zero but not stable rank one.\n\n\\begin{proof}[Proof of Theorem~\\ref{P_9Z26_XisCantor}]\nChoose any one point subset $Y \\S X$.\nSince ${\\mathrm{tsr}} (D) = 1$,\nProposition \\ref{TsrRR0A_Y}(\\ref{TsrRR0A_Y_tsr}) implies\nthat the subalgebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\n(see Definition~\\ref{D_4Y12_OrbSubalg})\nhas stable rank one.\nIf also ${\\mathrm{RR}} (D) = 0$,\nthen Proposition \\ref{TsrRR0A_Y}(\\ref{TsrRR0A_Y_RR}) implies\n${\\mathrm{tsr}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\big) = 1$.\nNow $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nby the appropriate part of Corollary \\ref{AyCentLargeConds}.\nSo ${\\mathrm{tsr}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\big) = 1$\nby Theorem 6.3 of~\\cite{ArPh},\nand if ${\\mathrm{RR}} (D) = 0$, then Theorem~6.4 of~\\cite{ArPh}\ngives ${\\mathrm{RR}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\big) = 0$.\n\\end{proof}\n\nWith more restrictive assumptions on~$D$,\nenough is known to get results when $\\dim (X) = 1$.\n\n\\begin{prp}\\label{P_9027_FibPrdT_sr1}\nLet $A$, $B$, and $C$ be C*-algebra{s},\nand let $\\varphi \\colon A \\to C$ and $\\psi \\colon B \\to C$ be homomorphism{s}.\nLet $D = A \\oplus_{C, \\varphi, \\psi} B$ (Definition~\\ref{pullback}).\nAssume that $\\psi$ is surjective,\nand that $A$ and $B$ have stable rank one.\nThen $D$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nSet $J = {\\operatorname{Ker}} (\\psi)$.\nWe have a commutative diagram with exact rows (maps defined below):\n\\[\n\\begin{CD}\n0 @>>> J @>{\\lambda}>> D @>{\\pi}>> A @>>> 0 \\\\\n& & @VV{{\\mathrm{id}}_B}V @VV{\\sigma}V @VV{\\gamma}V \\\\\n0 @>>> J @>{\\iota}>> B @>{\\kappa}>> B\/J @>>> 0.\n\\end{CD}\n\\]\nHere $\\iota$ is the inclusion of $J$ in~$B$\nand $\\kappa$ is the quotient map,\n$\\pi \\colon D \\to A$ and $\\sigma \\colon D \\to B$\nare the restrictions of the coordinate projections\n$(a, b) \\to a$ and $(a, b) \\to b$,\nand $\\lambda (b) = (b, 0)$ for $b \\in J$.\nFor $a \\in A$,\ndefine $\\gamma (a)$ by\nfirst choosing $b \\in B$ such that $\\psi (b) = \\varphi (a)$,\nand then setting $\\gamma (a) = \\kappa (b)$.\nIt is easy to check that $\\gamma$ is well defined,\nand that the diagram commutes.\nAll parts of exactness are immediate except that\nsurjectivity of $\\pi$ follows from\nsurjectivity of~$\\psi$.\n\nThe algebra $A$ has stable rank one by hypothesis,\nand $J$ has stable rank one by Theorem~4.4 of~\\cite{Rffl6}.\nNext,\n${\\mathrm{ind}} \\colon K_1 (B \/ J) \\to K_0 (J)$ is the zero map\nby Corollary~2 of~\\cite{Nag},\nso $K_0 (\\iota)$ is injective by the long exact sequence in\nK-theory for the bottom row.\nSince $K_0 (\\iota) = K_0 (\\sigma) \\circ K_0 (\\lambda)$,\nit follows that $K_0 (\\lambda)$ is injective.\nTherefore ${\\mathrm{ind}} \\colon K_1 (A) \\to K_0 (J)$ is the zero map\nby the long exact sequence in\nK-theory for the top row.\nNow $D$ has stable rank one by Corollary~2 of~\\cite{Nag}.\n\\end{proof}\n\nWe state for convenient reference the result on the stable rank\nof direct limits.\n\n\\begin{lem}\\label{L_0111_tsr_lim}\nLet $(A_{\\lambda})_{\\lambda \\in \\Lambda}$ be a direct system of C*-algebra{s}.\nThen\n${\\mathrm{tsr}} \\bigl( \\varinjlim_{\\lambda} A_{\\lambda} \\bigr)\n \\leq \\liminf_{\\lambda} {\\mathrm{tsr}} (A_{\\lambda})$.\n\\end{lem}\n\nWe do not assume that the maps of the direct system are injective.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_0111_tsr_lim}]\nWhen $\\Lambda = {\\mathbb{Z}}_{> 0}$,\nthis is Theorem~5.1 of~\\cite{Rffl6}.\nThe proof for general direct limits is the same.\n\\end{proof}\n\n\\begin{prp}\\label{P_9027_CXD_Tsr1}\nLet $D$ be a simple unital C*-algebra{}\nwith stable rank one and real rank zero,\nand suppose that $K_1 (D) = 0$.\nLet $X$ be a compact Hausdorff space{} with $\\dim (X) \\leq 1$.\nThen $C (X, D)$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nIf $X$ is a point, this is trivial.\nIf $X = [0, 1]$,\nit follows from Theorem~4.3 of~\\cite{NagOsPh2001}.\nIf $X$ is a one dimensional finite complex,\nthe result follows from these facts\nand Proposition~\\ref{P_9027_FibPrdT_sr1}\nby induction on the number of cells.\n\nNow suppose $X$ is a compact metric space.\nBy Corollary 5.2.6 of~\\cite{SkiK},\nthere is an inverse system $(X_n)_{n \\in {\\mathbb{Z}}_{\\geq 0}}$\nof one dimensional finite complexes\nsuch that $X \\cong \\varprojlim_n X_n$.\nThen $C (X, D) \\cong \\varinjlim_n C (X_n, D)$.\nSince $C (X_n, D)$ has stable rank one for all $n \\in {\\mathbb{Z}}_{\\geq 0}$,\nit follows from Lemma~\\ref{L_0111_tsr_lim}\nthat $C (X, D)$ has stable rank one.\n\nFinally,\nlet $X$ be a general compact Hausdorff space{} with $\\dim (X) \\leq 1$.\nBy Theorem~1 of~\\cite{Mdsc}\n(in Section~3 of that paper),\nthere is an inverse system $(X_{\\lambda})_{\\lambda \\in \\Lambda}$\nof one dimensional compact metric space{s}\nsuch that $X \\cong \\varprojlim_{\\lambda} X_{\\lambda}$.\nIt now follows that $C (X, D)$ has stable rank one\nby the same reasoning as in the previous paragraph.\n\\end{proof}\n\n\\begin{prp}\\label{P_9027_Rsha_Tsr1}\nLet $D$ be a simple unital C*-algebra.\nLet $R$ be a recursive subhomogeneous C*-algebra{} over~$D$\n(Definition~\\ref{RSHA})\nwith topological dimension at most~$1$\n(Definition \\ref{RSHAMachinery}(\\ref{9223_RSHAMachinery_6})).\nIf $D$ has stable rank one and real rank zero,\nand $K_1 (D) = 0$,\nthen $R$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nThe proof is the same as that of Proposition~\\ref{DStableRSHA},\nexcept using Proposition~\\ref{P_9027_FibPrdT_sr1}\nin place of Lemma~\\ref{DStablePullback},\nand using Proposition~\\ref{P_9027_CXD_Tsr1}.\n\\end{proof}\n\n\\begin{prp}\\label{P_9Z21_dim1}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $\\dim (X) \\leq 1$,\nthat $D$ has stable rank one and real rank zero,\nand that $K_1 (D) = 0$.\nLet $Y \\subset X$ be closed and nonempty.\nThen\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 1$.\n\\end{prp}\n\n\\begin{proof}\nFirst suppose that ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen the recursive subhomogeneous algebra over~$D$\nin Theorem~\\ref{AyRSHA}\nhas base spaces $Y_k$ which,\nfollowing Notation~\\ref{ModifiedRokhlinTower},\nare closed subsets of~$X$.\nBy Proposition 3.1.3 of~\\cite{Prs},\nthey therefore have covering dimension at most~$1$.\nThe result now follows from Proposition~\\ref{P_9027_Rsha_Tsr1}.\n\nStable rank one is preserved by direct limits\n(Lemma~\\ref{L_0111_tsr_lim}),\nso the general case now follows from\nProposition~\\ref{AyRSHADirLim}.\n\\end{proof}\n\nThe obvious examples of minimal homeomorphism{s}\nof one dimensional spaces are irrational rotations on~$S^1$,\nbut there are others.\nSee, for example, \\cite{GjdJhn},\nand the work on minimal homeomorphism{s} of the product of the Cantor set\nand the circle in \\cite{LnMti1} and~\\cite{LnMti3}.\n\n\\begin{thm}\\label{T_9222_Tsr1AndRR0_CrPrd}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $\\dim (X) \\leq 1$,\nthat $D$ has stable rank one and real rank zero,\nand that $K_1 (D) = 0$.\nFurther assume that one of the following conditions holds:\n\\begin{enumerate}\n\\item\\label{9927_AyCentLargeConds_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9927_AyCentLargeConds_SP}\nThe order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9927_AyCentLargeConds_CptGen}\nThe set\n$\\bigl\\{ \\alpha_x^n \\colon {\\mbox{$x \\in X$ and $n \\in {\\mathbb{Z}}$}} \\bigr\\}$\nis contained in a subset of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nhas stable rank one.\n\\end{thm}\n\n\\begin{proof}\nCombine Theorem 6.3 of~\\cite{ArPh},\nCorollary~\\ref{AyCentLargeConds},\nand Proposition~\\ref{P_9027_Rsha_Tsr1}.\n\\end{proof}\n\n\\section{Minimality of products of Denjoy homeomorphisms\n of the Cantor set}\\label{Sec_9929_MinProd}\n\n\\indent\nFor use in examples in the next section,\nwe prove minimality of products of Denjoy homeomorphisms\nwith rationally independent rotation numbers.\nWe have not found this result in the literature.\nThe method we use here is not the traditional approach\nto this kind of problem,\nbut we hope it will be useful elsewhere.\n\n\\begin{ntn}\\label{N_9X04_Rotation}\nFor $\\theta \\in {\\mathbb{R}}$ we let $r_{\\theta} \\colon S^1 \\to S^1$\ndenote the rotation by $2 \\pi \\theta$,\nthat is, $r_{\\theta} (\\zeta) = \\exp (2 \\pi i \\theta) \\zeta$\nfor $\\zeta \\in S^1$.\n\\end{ntn}\n\n\n\\begin{prp}\\label{P_4X04_ProdRotation}\nLet $I$ be a set,\nand let $(\\theta_i)_{i \\in I}$ be a family of elements of~${\\mathbb{R}}$.\nSuppose that $1$ and the numbers $\\theta_i$ for $i \\in I$ are\nlinearly independent over~${\\mathbb{Q}}$.\nThen the product $r \\colon (S^1)^{I} \\to (S^1)^{I}$\nof the rotations $r_{\\theta_i}$ is minimal.\n\\end{prp}\n\n\\begin{proof}\nIf $I$ is finite,\nthis is Proposition 1.4.1 of~\\cite{KtHsb}.\nThe general case follows from the finite case by taking the inverse\nlimit of the products over finite subsets of~$I$,\nby the discussion after Proposition II.4 of~\\cite{Frtg}.\nThe proof is is written for actions of the semigroup~${\\mathbb{Z}}_{\\geq 0}$\nand for countable inverse limits,\nbut the proof for actions of ${\\mathbb{Z}}$ and arbitrary inverse limits\nis the same.\n\\end{proof}\n\n\\begin{dfn}\\label{D_9929_StrOpen}\nLet $X$ and $Y$ be topological spaces,\nand let $g \\colon Y \\to X$.\nLet $y \\in Y$.\nWe say that $g$ is {\\emph{strictly open}} at~$y$\nif\n\\[\n\\bigl\\{ g^{-1} (U) \\colon\n {\\mbox{$U \\S X$ is open and $g (y) \\in U$}} \\bigr\\}\n\\]\nis a neighborhood base for $y$ in~$Y$.\n\\end{dfn}\n\nSome of the theory below works without assuming $g$ is continuous,\nso we do not require that neighborhood{s} be open.\n\nThe next lemma gives an alternate interpretation of strict openness.\n\n\\begin{lem}\\label{L_9929_WhenStrOpen}\nIn the situation in Definition~\\ref{D_9929_StrOpen},\nassume that $g$ is continuous{} and surjective, $Y$ is compact,\nand $X$ and $Y$ are Hausdorff.\nThen $g$ is strictly open at~$y$ if and only if{}\nthe following conditions hold:\n\\begin{enumerate}\n\\item\\label{Item_D_9929_Inj}\n$g^{-1} (\\{ g (y) \\}) = \\{ y \\}$.\n\\item\\label{Item_D_9929_Open}\nFor every open set $V \\S Y$ with $y \\in V$,\nthe set $g (V)$ is a neighborhood{} of $g (y)$.\n\\end{enumerate}\n\\end{lem}\n\nThe lemma is false without compactness of~$Y$.\nTake\n\\[\nX = [0, 1]\n\\qquad {\\mbox{and}} \\qquad\nY = \\{ (0, 0) \\} \\cup \\bigl( (0, 1]\\times [0, 1] \\bigr) \\S [0, 1]^2,\n\\]\nlet $g$ be projection to the first coordinate,\nand take $y = (0, 0)$.\nThen (\\ref{Item_D_9929_Inj}) and~(\\ref{Item_D_9929_Open}) hold,\nbut $g$ is not strictly open at~$y$.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_9929_WhenStrOpen}]\nAssume that $g$ is strictly open at~$y$.\nTo prove~(\\ref{Item_D_9929_Inj}),\nlet $z \\in Y \\setminus \\{ y \\}$,\nchoose $V \\S Y$ open with $y \\in V$ and $z \\not\\in V$,\nand choose $U \\S X$ open such that $g (y) \\in U$\nand $g^{-1} (U) \\S V$.\nThen $z \\not\\in g^{-1} (U)$, so $g (z) \\neq g (y)$.\n\nTo prove~(\\ref{Item_D_9929_Open}),\nlet $V \\S Y$ be open with $y \\in V$.\nChoose $U \\S X$ open such that $g (y) \\in U$\nand $g^{-1} (U) \\S V$.\nSurjectivity of~$g$ implies $U \\S g ( g^{-1} (U) ) \\S g (V)$,\nso $g (V)$ is a neighborhood{} of $g (y)$.\n\nFor the converse,\nassume~(\\ref{Item_D_9929_Open}),\nand suppose that $g$ is not strictly open at~$y$.\nLet ${\\mathcal{N}}$ be the set of open subsets of~$X$ which contain $g (y)$,\nordered by reverse inclusion.\nSince $g$ is continuous,\nfailure of strict openness means we can\nchoose $V \\S Y$ open with $y \\in V$\nand such that for every $U \\in {\\mathcal{N}}$,\nwe have $g^{-1} (U) \\not\\S V$.\nSo there is $z_U \\in Y$ such that $z_U \\not\\in V$ but $g (z_U) \\in U$.\nThe net $(z_U)_{U \\in {\\mathcal{N}}}$\nsatisfies $g (z_U) \\to g (y)$.\nChoose a subnet $(t_{\\lambda})_{\\lambda \\in \\Lambda}$\nsuch that $t = \\lim_{\\lambda} t_{\\lambda}$ exists.\nThen $t \\not\\in V$.\nAlso $g (t_{\\lambda}) \\to g (t)$ by continuity and $g (t_{\\lambda}) \\to g (y)$\nby construction, so $g (t) = g (y)$.\nWe have contradicted~(\\ref{Item_D_9929_Inj}).\n\\end{proof}\n\n\\begin{lem}\\label{L_9929_ExtMin}\nLet $X$ and $Y$ be compact Hausdorff space{s},\nand let $h \\colon X \\to X$ and $k \\colon Y \\to Y$ be homeomorphism{s}.\nLet $g \\colon Y \\to X$ be continuous and surjective,\nand satisfy $g \\circ k = h \\circ g$.\nSuppose that $h$ is minimal and there is a dense subset $S \\S Y$\nsuch that $g$ is strictly open at every point of~$S$.\nThen $k$ is minimal.\n\\end{lem}\n\n\\begin{proof}\nIt is enough to show that for every $y \\in Y$ and every\nnonempty open set $V \\S Y$,\nthere is $n \\in {\\mathbb{Z}}$ such that $k^n (y) \\in V$.\nChoose $z \\in S \\cap V$.\nSince $g$ is strictly open at~$z$,\nthere is an open set $U \\S X$ such that $g (z) \\in U$\nand $g^{-1} (U) \\S V$.\nSince $h$ is minimal,\nthere is $n \\in {\\mathbb{Z}}$ such that $h^n (g (y)) \\in U$.\nThen $g (k^n (y)) \\in U$,\nso $k^n (y) \\in g^{-1} (U) \\S V$.\n\\end{proof}\n\nIt is well known that Lemma~\\ref{L_9929_ExtMin} fails\nwithout strict openness,\neven if one assumes that $g^{-1} (x)$ is finite for all $x \\in X$\nand has only one point for a dense subset of points $x \\in X$.\nThe following example was suggested by B.~Weiss.\n\n\\begin{exa}\\label{E_9X04_NeedStrict}\nFix $\\theta \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$.\nTake $X = S^1$ and take $h = r_{\\theta}$\n(rotation by $2 \\pi \\theta$; Notation~\\ref{N_9X04_Rotation}).\nDefine $Y \\S S^1 \\times [0, 1]$ to be\n\\[\nY = \\bigl( S^1 \\times \\{ 0 \\} \\bigr)\n \\cup \\left\\{ \\left( e^{2 \\pi i n \\theta}, \\, \\frac{1}{| n | + 1} \\right)\n \\colon n \\in {\\mathbb{Z}} \\right\\},\n\\]\nand define $k \\colon Y \\to Y$ to be\n$k (\\zeta, 0) = (r_{\\theta} (\\zeta), \\, 0)$ for $\\zeta \\in S^1$ and\n\\[\nk \\left( e^{2 \\pi i n \\theta}, \\, \\frac{1}{| n | + 1} \\right)\n = \\left( e^{2 \\pi i (n + 1) \\theta}, \\, \\frac{1}{| n + 1 | + 1} \\right)\n\\]\nfor $n \\in {\\mathbb{Z}}$.\nLet $g \\colon Y \\to X$ be projection to the first coordinate.\nThen $g \\circ k = h \\circ g$,\n$g$ is surjective,\n$h$ is minimal,\n$k$ is not minimal,\n$g^{-1} (x)$ has at most two points for every $x \\in X$,\nand $g^{-1} (x)$ has one point for all but countably many $x \\in X$.\n\\end{exa}\n\nThe following two easy lemmas will be used in the construction of\nexamples.\n\n\\begin{lem}\\label{L_9X04_Prod}\nLet $I$ be a nonempty set,\nand for $i \\in I$ let $g_i \\colon X_i \\to Y_i$\nand $y_i \\in Y_i$ be as in Definition~\\ref{D_9929_StrOpen},\nwith $g_i$ strictly open at~$y_i$.\nSet\n\\[\nX = \\prod_{i \\in I} X_i\n\\qquad {\\mbox{and}} \\qquad\nY = \\prod_{i \\in I} Y_i,\n\\]\nlet $g \\colon Y \\to X$ be given by $g (z) = ( g_i (z_i))_{i \\in I}$\nfor $z = (z_i)_{i \\in I} \\in Y$,\nand set $y = (y_i)_{i \\in I}$.\nThen $g$ is strictly open at~$y$.\n\\end{lem}\n\n\\begin{proof}\nIt is obvious that $g^{-1} ( \\{ g (y) \\} ) = \\{ y \\}$.\nNow let $V \\S Y$ be open with $y \\in V$.\nWe need an open set $U \\S X$\nsuch that $g (y) \\in U$, $g^{-1} (U)$ is a neighborhood{} of~$y$,\nand $g^{-1} (U) \\S V$.\nWithout loss of generality{} there are open subsets $V_i \\S Y_i$\nand a finite set $F \\S I$\nsuch that $V = \\prod_{i \\in I} V_i$,\n$V_i = Y_i$ for all $i \\in I \\setminus F$,\nand $y_i \\in V_i$ for all $i \\in I$.\nFor $i \\in F$ choose an open set $U_i \\S X_i$ with $g_i (y_i) \\in U_i$\nand such that $g_i^{-1} (U_i) \\S V_i$.\nSet $U_i = X_i$ for $i \\in I \\setminus F$.\nSince $g_i^{-1} (U_i)$ is a neighborhood{} of~$y_i$\nfor all $i \\in I$\nand $U_i = X_i$ for $i \\in I \\setminus F$,\nthe set $g^{- 1} (U) = \\prod_{i \\in I} g_i^{-1} (U_i)$\nis a neighborhood{} of~$y$.\nSo the set $U = \\prod_{i \\in I} U_i$ is the required set.\n\\end{proof}\n\n\\begin{lem}\\label{L_9X04_Subsp}\nIn the situation in Definition~\\ref{D_9929_StrOpen},\nassume that $g$ is strictly open at~$y$.\nLet $Y_0 \\S Y$ be a subspace such that $y \\in Y_0$.\nThen $g |_{Y_0}$ is strictly open at~$y$.\n\\end{lem}\n\n\\begin{proof}\nThe result is immediate from the definition of the subspace\ntopology.\n\\end{proof}\n\n\\begin{lem}\\label{L_9929_S1toS1}\nLet $g \\colon S^1 \\to S^1$ be continuous{} and surjective.\nSuppose that there are disjoint closed arcs\n$I_1, I_2, \\ldots \\S S^1$\nsuch that:\n\\begin{enumerate}\n\\item\\label{Item_L_9929_S1toS1_Const}\nFor $n \\in {\\mathbb{Z}}_{> 0}$, the function $g$ is constant on~$I_n$.\n\\item\\label{Item_L_9929_S1toS1_NI}\n$\\bigcup_{n = 1}^{\\infty} I_n$ is dense in~$S^1$.\n\\item\\label{Item_L_9929_S1toS1_Inj}\nWith $R = S^1 \\setminus \\bigcup_{n = 1}^{\\infty} I_n$,\nthe restriction $g |_R$ is injective.\n\\end{enumerate}\nThen $g$ is strictly open at every point of~$R$.\n\\end{lem}\n\n\\begin{proof}\nFor distinct $\\lambda, \\zeta \\in S^1$,\nwe denote the open, closed, and half open arcs from $\\lambda$ to~$\\zeta$\nusing interval notation:\n$(\\lambda, \\zeta)$, $[\\lambda, \\zeta]$, $[\\lambda, \\zeta)$, and $(\\lambda, \\zeta]$.\nFurther write $I_n = [\\beta_n, \\gamma_n]$ with $\\beta_n, \\gamma_n \\in S^1$,\nand set $B = \\{ \\beta_n \\colon n \\in {\\mathbb{Z}}_{> 0} \\}$\nand $C = \\{ \\gamma_n \\colon n \\in {\\mathbb{Z}}_{> 0} \\}$.\n\nThe set $g (S^1 \\setminus R)$ is countable\nand $g$ is surjective, so $S^1 \\setminus g (R)$ is countable.\n\nFirst,\nwe claim that if $E, F \\S S^1$ are disjoint,\nthen ${\\mathrm{int}} (g (E)) \\cap {\\mathrm{int}} (g (F)) = \\varnothing$.\nIf the claim is false,\nthen $g (E) \\cap g (F)$ is uncountable.\nNow $g (E) \\setminus g (E \\cap R) \\S g (S^1 \\setminus R)$\nwhich is countable, so $g (E) \\setminus g (E \\cap R)$ is countable.\nSimilarly $g (F) \\setminus g (F \\cap R)$ is countable.\nSo $g (E \\cap R) \\cap g (F \\cap R)$ is uncountable.\nThis contradicts $E \\cap F = \\varnothing$ and injectivity of $g |_R$.\n\nSecond, we claim that if $J \\S S^1$ is any nonempty open arc such that\nthere is no $n \\in {\\mathbb{Z}}_{> 0}$ with ${\\overline{J}} \\S I_n$,\nthen $R \\cap J \\neq \\varnothing$.\nSuppose the claim is false.\nWrite $J = (\\lambda, \\zeta)$ with $\\lambda, \\zeta \\in S^1$.\nDefine\n\\[\nS = \\begin{cases}\n \\varnothing & \\hspace*{1em} \\lambda \\not\\in R\n \\\\\n \\{ \\lambda \\} & \\hspace*{1em} \\lambda \\in R\n\\end{cases}\n\\qquad {\\mbox{and}} \\qquad\nT = \\begin{cases}\n \\varnothing & \\hspace*{1em} \\zeta \\not\\in R\n \\\\\n \\{ \\zeta \\} & \\hspace*{1em} \\zeta \\in R.\n\\end{cases}\n\\]\nThen ${\\overline{J}}$ is the disjoint union of the closed sets\n\\[\nS, \\, T, \\, {\\overline{J}} \\cap I_1, \\, {\\overline{J}} \\cap I_2, \\ldots.\n\\]\nSuppose there is $n \\in {\\mathbb{Z}}_{> 0}$ such that ${\\overline{J}} \\cap I_m = \\varnothing$\nfor all $m \\neq n$.\nSince ${\\overline{J}} \\not\\subset I_n$,\nthere is a nonempty open arc $L \\S J$\nsuch that $L \\cap I_n = \\varnothing$,\nwhence $L \\cap I_m = \\varnothing$ for all $m \\in {\\mathbb{Z}}_{> 0}$.\nThis contradicts~(\\ref{Item_L_9929_S1toS1_NI}).\nThus, at least two of\nthe sets ${\\overline{J}} \\cap I_n$ are nonempty.\nBut, according to\nTheorem~6 in Part III of Section~47 (in Chapter~5) of~\\cite{Krtw},\na connected compact Hausdorff space{} cannot be the disjoint union of closed subsets\n$E_1, E_2, \\ldots$ with at least two of them nonempty.\nThis contradiction proves the claim.\n\nThird, we claim that, under the same hypotheses, $R \\cap J$\nis infinite.\nAgain write $J = (\\lambda, \\zeta)$ with $\\lambda, \\zeta \\in S^1$.\nApply the previous claim\nto choose $\\rho_1 \\in R \\cap J$ and set $J_1 = (\\rho_1, \\zeta)$.\nThen $\\rho_1 \\not\\in \\bigcup_{n = 1}^{\\infty} I_n$\nso there is no $n \\in {\\mathbb{Z}}_{> 0}$ with ${\\overline{J_1}} \\S I_n$.\nApply the previous claim again, choosing $\\rho_2 \\in R \\cap J_1$,\nand showing that $J_2 = (\\rho_2, \\zeta)$\nsatisfies ${\\overline{J_1}} \\not\\S I_n$ for all $n \\in {\\mathbb{Z}}_{> 0}$.\nProceed inductively.\n\nFourth, we claim that if $\\lambda \\in R \\cup C$ and $\\zeta \\in R \\cup B$\nare distinct,\nthen $g \\bigl( (\\lambda, \\zeta) \\bigr)$ is an open arc.\nFor the proof, set $J = (\\lambda, \\zeta)$.\nWe can assume that $g (J) \\neq S^1$.\nSince $g (J)$ is connected, there are $\\mu, \\nu \\in S^1$\nsuch that $g (J)$ is one of\n$(\\mu, \\nu)$, $[\\mu, \\nu]$\n(in this case we allow $\\mu = \\nu$), $[\\mu, \\nu)$, or $(\\mu, \\nu]$.\nSuppose $\\mu \\in g (J)$; we will obtain a contradiction.\nChoose $\\rho \\in J$ such that $g (\\rho) = \\mu$.\nThere are two cases: $\\rho \\in R$ and $\\rho \\in \\bigcup_{n = 1}^{\\infty} I_n$.\n\nIn the first case, let $L_0$ and $L_1$ be the nonempty arcs\n$L_0 = (\\lambda, \\rho)$ and $L_1 = (\\rho, \\zeta)$.\nFor $j = 0, 1$,\nsince $\\rho \\in {\\overline{L_j}}$,\nthe arc $L_j$ satisfies the hypotheses of the second claim.\nSo $R \\cap L_j$ is infinite by the third claim.\nSince $g |_R$ is injective, $g (L_j)$ is infinite.\nClearly $\\mu \\in {\\overline{g (L_j)}} \\S [\\mu, \\nu]$.\nAlso, $g (L_j)$ is an arc by connectedness.\nSo there is $\\omega_j \\in (\\mu, \\nu)$ such that $(\\mu, \\omega_j) \\S g (L_j)$.\nThis implies ${\\mathrm{int}} ( g (L_0) ) \\cap {\\mathrm{int}} ( g (L_1) ) \\neq \\varnothing$,\ncontradicting the first claim.\n\nIn the second case,\nlet $n \\in {\\mathbb{Z}}_{> 0}$ be the integer such that $\\rho \\in I_n$.\nBoth $\\lambda \\in [\\beta_n, \\gamma_n)$ and $\\zeta \\in (\\beta_n, \\gamma_n]$\nare impossible, but $[\\beta_n, \\gamma_n] \\cap (\\lambda, \\zeta) \\neq \\varnothing$,\nso $[\\beta_n, \\gamma_n] \\subset (\\lambda, \\zeta)$.\nTherefore the arcs $L_0 = (\\lambda, \\beta_n)$ and $L_1 = (\\gamma_n, \\zeta)$\nare nonempty and do not intersect~$I_n$.\nIf there is $m \\in {\\mathbb{Z}}_{> 0}$ such that ${\\overline{L_0}} \\S I_m$,\nthen clearly $m \\neq n$,\nbut then $\\beta_n \\in I_m \\cap I_n$,\ncontradicting $I_m \\cap I_n = \\varnothing$.\nSo no such $m$ exists,\nwhence $L_0$ satisfies the hypotheses of the second claim.\nSimilarly $L_1$ satisfies the hypotheses of the second claim.\nSince $g$ is constant on~$I_n$,\nwe have $g (\\beta_n) = g (\\gamma_n) = g (\\rho) = \\mu$.\nNow we get a contradiction in the same way as in the first case.\n\nSo $\\mu \\not\\in g (J)$.\nSimilarly $\\nu \\not\\in g (J)$.\nThe claim is proved.\n\nFifth,\nwe claim that if $L \\S S^1$ is an open arc,\nthen $g^{-1} (L)$ is an open arc.\nTo prove the claim,\nsince $g^{-1} (L)$ is open,\nthere are an index set $S$\nand disjoint nonempty open arcs $J_s$ for $s \\in S$\nsuch that $g^{-1} (L) = \\bigcup_{s \\in S} J_s$,\nand there are $\\lambda_s, \\zeta_s \\in S^1$\nsuch that $J_s = (\\lambda_s, \\zeta_s)$.\n\nLet $s \\in S$.\nSuppose $\\lambda_s \\in I_n$ for some $n \\in {\\mathbb{Z}}_{> 0}$.\nSince $g$ is constant on~$I_n$,\neither $I_n \\cap g^{-1} (L) = \\varnothing$\nor $I_n \\S g^{-1} (L)$.\nIn the second case, $I_n \\cup (\\lambda_s, \\zeta_s)$\nis a connected subset of $g^{-1} (L)$.\nSince $(\\lambda_s, \\zeta_s)$ is a maximal connected subset of $g^{-1} (L)$,\nwe have $I_n \\S (\\lambda_s, \\zeta_s)$.\nThis contradicts $\\lambda_s \\in I_n$.\nSo the first case must apply, and therefore $\\lambda_s = \\gamma_n$.\nCombining this with the possibility\n$\\lambda_s \\not\\in \\bigcup_{n = 1}^{\\infty} I_n$,\nwe conclude that $\\lambda_s \\in R \\cup C$.\nSimilarly, $\\zeta_s \\in R \\cup B$.\nBy the previous claim,\n$g \\bigl( (\\lambda_s, \\zeta_s) \\bigr)$ is open.\n\nThe first claim now implies that the sets\n$g \\bigl( (\\lambda_s, \\zeta_s) \\bigr)$ are all disjoint.\nSince $L$ is connected,\nit follows that ${\\mathrm{card}} (S) = 1$,\nwhich implies the claim.\n\nTo prove the lemma,\nlet $\\eta \\in R$ and let $V \\S S^1$ be open with $\\eta \\in V$.\nWe need an open set $U \\S S^1$\nsuch that $g (\\eta) \\in U$ and $g^{-1} (U) \\S V$.\nChoose $\\lambda_0, \\zeta_0 \\in S^1$\nsuch that $\\eta \\in (\\lambda_0, \\zeta_0) \\S V$.\nUse the second claim to choose $\\lambda_1 \\in (\\lambda_0, \\eta) \\cap R$\nand $\\zeta_1 \\in (\\eta, \\zeta_0) \\cap R$.\nThen $U = g \\bigl( (\\lambda_1, \\zeta_1) \\bigr)$ is open by\nthe fourth claim,\nand by the fifth claim there are $\\lambda_2, \\zeta_2 \\in S^1$\nsuch that $g^{-1} (U) = (\\lambda_2, \\zeta_2)$.\nIf $\\lambda_1 \\in g^{-1} (U)$,\nthen $(\\lambda_2, \\lambda_1)$ is a nonempty open arc contained in $g^{-1} (U)$\nwith $\\lambda_1 \\in R$.\nThe third claim and injectivity of $g |_R$\nimply that $g \\bigl( (\\lambda_2, \\lambda_1) \\bigr)$\nis not a point.\nSince $g \\bigl( (\\lambda_2, \\lambda_1) \\bigr)$ is connected,\nit has nonempty interior.\nIt is contained in $U = g \\bigl( (\\lambda_1, \\zeta_1) \\bigr)$,\ncontradicting the first claim.\nSo $\\lambda_1 \\not\\in (\\lambda_2, \\zeta_2)$.\nSimilarly $\\zeta_1 \\not\\in (\\lambda_2, \\zeta_2)$.\nSince $\\eta \\in (\\lambda_2, \\zeta_2)$,\n$\\eta \\in (\\lambda_1, \\zeta_1)$, and $(\\lambda_2, \\zeta_2)$ is connected,\nit now follows that $(\\lambda_2, \\zeta_2) \\S (\\lambda_1, \\zeta_1)$.\n(In fact, we have equality, since $g \\bigl( (\\lambda_1, \\zeta_1) \\bigr) = U$.)\nThus\n\\[\ng^{-1} (U)\n = (\\lambda_2, \\zeta_2)\n \\S (\\lambda_1, \\zeta_1)\n \\S (\\lambda_0, \\zeta_0)\n \\S V.\n\\]\nThis completes the proof.\n\\end{proof}\n\nWe have not seen a term in the literature for the following\nclass of homeomorphism{s}.\n\n\\begin{dfn}\\label{D_9X04_RestrDj}\nA {\\emph{restricted Denjoy homeomorphism}}\nis a homeomorphism of the Cantor set which is conjugate to\nthe restriction and corestriction of a Denjoy homeomorphism of~$S^1$,\nin the sense of Definition~3.3 of~\\cite{PtSmdSk},\nto the unique minimal set of Proposition~3.4 of~\\cite{PtSmdSk}.\nThe {\\emph{rotation number}}\nof a restricted Denjoy homeomorphism is the rotation number,\nas at the beginning of Section~3 of~\\cite{PtSmdSk},\nof the Denjoy homeomorphism of~$S^1$\nof which it is the restriction;\nthis number is well defined by Remark~3 at the end of Section~3\nof~\\cite{PtSmdSk}.\n\\end{dfn}\n\n\\begin{lem}\\label{L_9X04_DjSemiCj}\nLet $X$ be the Cantor set,\nand let $k \\colon X \\to X$ be a restricted Denjoy homeomorphism\nwith rotation number~$\\theta$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nThen there are a continuous{} surjective map $g \\colon X \\to S^1$\nand a dense subset $R \\S X$ such that\n$g$ is strictly open at~$x$ for every $x \\in R$\nand, following Notation~\\ref{N_9X04_Rotation},\n$g \\circ h = r_{\\theta} \\circ g$.\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $k_0 \\colon S^1 \\to S^1$ is a Denjoy homeomorphism\nas in Definition~3.3 of~\\cite{PtSmdSk},\nthat $X \\S S^1$ is the unique minimal set for~$k_0$,\nand that $k$ is the restriction and corestriction of $k_0$ to~$X$.\nCorollary~3.2 of~\\cite{PtSmdSk}\ngives $h \\colon S^1 \\to S^1$\nsuch that $h \\circ k_0 = r_{\\theta} \\circ h$.\nSet $g = h |_X$.\nSince $X \\S S^1$ is compact and invariant and $r_{\\theta}$ is minimal,\n$g$ must be surjective.\nThe discussion after Proposition~3.4 of~\\cite{PtSmdSk}\nimplies that $h$ satisfies the hypotheses of Lemma~\\ref{L_9929_S1toS1},\nwith $X = S^1 \\setminus \\bigcup_{n = 1}^{\\infty} {\\mathrm{int}} (I_n)$.\nMoreover, the set $R$ of Lemma~\\ref{L_9929_S1toS1}\nis nonempty by\nTheorem~6 in Part III of Section~47 (in Chapter~5) of~\\cite{Krtw},\nand $k$-invariant,\nhence dense in~$X$.\nBy Lemma~\\ref{L_9929_S1toS1} and Lemma~\\ref{L_9X04_Subsp},\n$g$ is strictly open at every point of~$R$.\n\\end{proof}\n\n\\begin{prp}\\label{P_4X04_ProdRstDj}\nLet $I$ be a set,\nlet $(\\theta_i)_{i \\in I}$ be a family of elements of~${\\mathbb{R}}$,\nand for $i \\in I$ let $k_i \\colon X_i \\to X_i$\nbe a restricted Denjoy homeomorphism with rotation number~$\\theta_i$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nSuppose that $1$ and the numbers $\\theta_i$ for $i \\in I$ are\nlinearly independent over~${\\mathbb{Q}}$.\nThen the product $k \\colon \\prod_{i \\in I} X_i \\to \\prod_{i \\in I} X_i$\nof the homeomorphism{s} $k_i$ is minimal.\n\\end{prp}\n\n\\begin{proof}\nFor $i \\in I$,\nlet $g_i \\colon X_i \\to S^1$ be as in Lemma~\\ref{L_9X04_DjSemiCj}.\nBy Lemma~\\ref{L_9X04_DjSemiCj},\nthere is a dense subset $R_i \\S X_i$\nsuch that $g_i$ is strictly open at every point of~$R_i$.\nLet $g \\colon \\prod_{i \\in I} X_i \\to (S^1)^{I}$\nbe the product of the maps~$g_i$.\nLemma~\\ref{L_9X04_Prod} implies that\n$g$ is strictly open at every point of $\\prod_{i \\in I} R_i$,\nthis set is dense,\nand the product $r \\colon (S^1)^{I} \\to (S^1)^{I}$\nof the rotations $r_{\\theta_i}$ is minimal\nby Proposition~\\ref{P_4X04_ProdRotation}.\nSo $k$ is minimal\nby Lemma~\\ref{L_9929_ExtMin}.\n\\end{proof}\n\n\n\\section{Examples}\\label{Sec_InfRed}\n\n\\indent\nTo take full advantage of Proposition~\\ref{AyRSHADirLim},\nwe need information on the structure of simple\ndirect limits of recursive subhomogeneous algebras\nover the algebra $D$ which appears there.\nFor example,\nwe need conditions for such direct limits to have stable rank~$1$\n(as originally done for $D = {\\mathbb{C}}$ in~\\cite{DNNP})\nand to have real rank zero\n(as originally done for $D = {\\mathbb{C}}$ in~\\cite{BDR}).\nWe intend to study this problem in a future paper.\nEven without generalizations of those theorems,\nthree cases are accessible now.\nThey are $h \\colon X \\to X$ wild\n(for example, with strictly positive mean dimension,\nas in Notation~\\ref{N_0108_DimX})\nbut with $D$ being ${\\mathcal{Z}}$-stable;\n$X$ is the Cantor set\nand $D$ has stable rank one but is otherwise wild;\nand $X = S^1$,\n$h$ is an irrational rotation,\nand $D$ has stable rank one, real rank zero,\nand trivial $K_1$-group,\nbut is otherwise wild.\n\nWe also give examples for Theorem~\\ref{T_0101_PICase},\nalthough only for $G = {\\mathbb{Z}}$.\n\nIn the first type of example,\nif $X$ is finite dimensional{} then the action has a\nhigher dimensional Rokhlin property,\nand results on crossed products by such actions can be applied.\nAs far as we know, however,\nthere are no previously known general theorems\nwhich apply when $X$ is infinite dimensional.\nIn the second type of example,\nthe action has the Rokhlin property.\nSince the algebra is not simple,\nresults of~\\cite{OP1} can't be applied,\nand, when $D$ is not ${\\mathcal{Z}}$-stable\nand does not have finite nuclear dimension,\nTheorems 4.1 and~5.8 of~\\cite{HWZ} can't be applied.\nWe know of no previous general theorems which apply in this case.\nActions in the third type of example\nare further away from known results:\nthe situation is like for the second type of example,\nbut the Rokhlin property must be weakened to\nfinite Rokhlin dimension with commuting towers.\n\nSince we will use quasifree automorphisms of reduced C*-algebras\nof free groups\nand the free shift on $C^*_{\\mathrm{r}} (F_{\\infty})$\nin several examples,\nwe establish notation for them separately.\n\n\\begin{ntn}\\label{N_9214_QFreeFn}\nFor $n \\in {\\mathbb{Z}}_{> 0} \\cup \\{ \\infty \\}$,\nwe let $F_n$ denote the free group on $n$~generators.\nWe let $u_1, u_2, \\ldots, u_n \\in C^*_{\\mathrm{r}} (F_n)$\n(when $n = \\infty$,\nfor $u_1, u_2, \\ldots \\in C^*_{\\mathrm{r}} (F_{\\infty})$)\nbe the ``standard'' unitaries in $C^*_{\\mathrm{r}} (F_n)$,\nobtained as the images of the standard generators of~$F_n$.\nFor $\\zeta = (\\zeta_1, \\zeta_2, \\ldots, \\zeta_n) \\in (S^1)^n$\n(when $n = \\infty$,\nfor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}_{> 0}} \\in (S^1)^{{\\mathbb{Z}}_{> 0}}$),\nwe let $\\varphi_{\\zeta} \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_n) )$\nbe the (quasifree) automorphism\ndetermined by $\\varphi_{\\zeta} (u_k) = \\zeta_k u_k$\nfor $k = 1, 2, \\ldots, n$\n(when $n = \\infty$,\nfor $k = 1, 2, \\ldots)$.\nIt is well known that $\\zeta \\mapsto \\varphi_{\\zeta}$ is continuous.\n\\end{ntn}\n\n\\begin{ntn}\\label{N_9214_FreeShift}\nTake the standard generators of the free group $F_{\\infty}$\nto be indexed by~${\\mathbb{Z}}$,\nand for $n \\in {\\mathbb{Z}}$ let $u_n \\in C^*_{\\mathrm{r}} (F_{\\infty})$\nbe the unitary obtained as the image of the corresponding\ngenerator of~$F_{\\infty}$.\nWe denote by $\\sigma$ the free shift on $C^*_{\\mathrm{r}} (F_{\\infty})$,\nthat is, the automorphism\n$\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\ndetermined by $\\sigma (u_n) = u_{n + 1}$\nfor $n \\in {\\mathbb{Z}}$.\n\\end{ntn}\n\n\\begin{exa}\\label{E_9214_ShiftAndUHF}\nSet $X_0 = (S^1)^{{\\mathbb{Z}}}$.\nLet $h_0 \\colon X_0 \\to X_0$\nbe the (forwards) shift,\ndefined for $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X_0$ by\n$h_0 (\\zeta) = (\\zeta_{n - 1})_{n \\in {\\mathbb{Z}}}$.\nLet $D = \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}} M_2$\nbe the $2^{\\infty}$~UHF algebra.\nChoose a bijection $\\sigma \\colon {\\mathbb{Z}}_{> 0} \\to {\\mathbb{Z}}$,\nand for $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X_0$\ndefine\n\\[\n\\alpha_{\\zeta}^{(0)}\n = \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}}\n {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n 1 & 0 \\\\\n 0 & \\zeta_{\\sigma (n)}\n \\end{matrix} \\right) \\right)\n \\in {\\mathrm{Aut}} (D).\n\\]\nThen $\\zeta \\mapsto \\alpha_{\\zeta}^{(0)}$ is continuous.\n\nWe have $\\mathrm{mdim} (h_0) = 1$ (see Notation~\\ref{N_0108_DimX})\nby Proposition~3.3 of~\\cite{LndWs},\nand we can use $X_0$, $h_0$, $D$, and $\\zeta \\mapsto \\alpha^{(0)}_{\\zeta}$\nin Lemma~\\ref{L_4Y12_GetAuto}.\nHowever, $h_0$ is not minimal.\nWe therefore proceed as follows.\nIdentify $[0, 1]$ with a closed arc in~$S^1$,\nsay via $\\lambda \\mapsto \\exp (\\pi i \\lambda)$.\nUse this identification to identify\n$[0, 1]^{{\\mathbb{Z}}}$ with a closed subset of $X_0 = (S^1)^{{\\mathbb{Z}}}$.\nThis identification is equivariant when both spaces\nare equipped with the shift homeomorphism{s}.\nLet $X \\S X_0$ and $h = h_0 |_{X} \\colon X \\to X$\nbe the minimal subshift\nin~\\cite{GlKr},\nwhich can be taken to have mean dimension\narbitrarily close to~$1$.\nFor $\\zeta \\in X$ let $\\alpha_{\\zeta} = \\alpha_{\\zeta}^{(0)}$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThen $\\alpha$ lies over the free minimal action of ${\\mathbb{Z}}$ on~$X$\ngenerated by~$h$.\n\nThe crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is simple\nby Proposition~\\ref{P_4Y15_CPSimple},\nand is nuclear,\nso it is ${\\mathcal{Z}}$-stable by Theorem~\\ref{ATraciallyZStable}.\n\\end{exa}\n\nThe action in Example~\\ref{E_9214_ShiftAndUHF}\ncan also be described as follows.\nRealize $D$ as the algebra of the canonical anticommutation\nrelations on generators $a_k$ for $k \\in {\\mathbb{Z}}_{> 0}$,\nfollowing Section~5.1 of~\\cite{Brtl}.\nThen $\\alpha_{\\zeta}^{(0)}$ is the gauge automorphism\n$\\alpha_{\\zeta}^{(0)} (a_k) = \\zeta_{\\sigma (k)} a_k$.\nIn~\\cite{Brtl}, see Section~5.1 and the proof of Lemma~5.2.\n\nThe next example is a slightly different version,\nwith larger mean dimension.\n\n\\begin{exa}\\label{E_9Z26_ShiftAndUHF_v2}\nLet $X_0$, $h_0$, $D$, and $\\zeta \\mapsto \\alpha_{\\zeta}^{(0)}$\nbe as in Example~\\ref{E_9214_ShiftAndUHF}.\nWe will, however, use the homeomorphism{} $h_0^2$ of~$X_0$,\nwhich is the shift on $(S^1 \\times S^1)^{{\\mathbb{Z}}}$.\nLet $(X, h)$ be the minimal subspace of the shift on $([0, 1]^2)^{{\\mathbb{Z}}}$\nconstructed in Proposition 3.5 of \\cite{LndWs},\nwhich satisfies $\\mathrm{mdim} (h) > 1$ (Notation~\\ref{N_0108_DimX}).\nUse an embedding of $[0, 1]^2$ in~$(S^1)^2$\nto choose an equivariant homeomorphism{} $g$\nfrom $(X, h)$ to an invariant closed subset of $(X_0, h_0^2)$.\nFor $x \\in X$ let $\\alpha_{x} = \\alpha_{g (x)}^{(0)}$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThe crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is again\nsimple by Proposition~\\ref{P_4Y15_CPSimple},\nso ${\\mathcal{Z}}$-stable by Theorem~\\ref{ATraciallyZStable}.\n\\end{exa}\n\n\\begin{exa}\\label{E_9214_ShiftAndCStFI_2}\nLet $X$, $h$, $D$, and $\\zeta \\mapsto \\alpha_{\\zeta}$ be as in\nExample~\\ref{E_9214_ShiftAndUHF}.\nDefine $E = D \\otimes C^*_{\\mathrm{r}} (F_{\\infty})$,\nand, following Notation~\\ref{N_9214_QFreeFn},\nfor $\\zeta \\in X$ define\n$\\beta_{\\zeta} = \\alpha_{\\zeta} \\otimes \\varphi_{\\zeta}$.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action $\\beta \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, E))$\nwhich lies over the free minimal action of ${\\mathbb{Z}}$ on~$X$\ngenerated by~$h$.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ is tracially ${\\mathcal{Z}}$-stable\nand has stable rank one.\nTracial ${\\mathcal{Z}}$-stability follows from Theorem~\\ref{ATraciallyZStable}.\nFor stable rank one,\n$E$ is exact,\nso has strict comparison of positive elements\nby Corollary 4.6 of~\\cite{Rdm7}.\nChoose any one point subset $Y \\S X$.\nThen $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)_Y$\n(see Definition~\\ref{D_4Y12_OrbSubalg})\nis ${\\mathcal{Z}}$-stable\nby Corollary~\\ref{EStableGivesZStable}.\nThis algebra\nis simple by Proposition~\\ref{P_4Y15_CPSimple},\nso by Theorem 6.7 of~\\cite{Rdm7} it has stable rank one.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)_Y$\nis centrally large\nin $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$\nby\nCorollary \\ref{AyCentLargeConds}(\\ref{9212_AyCentLargeConds_StrCmp}),\nso Theorem 6.3 of~\\cite{ArPh} implies\nthat $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ has stable rank one.\n\\end{exa}\n\nIn Example~\\ref{E_9214_ShiftAndCStFI_2}, we don't know whether\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ is ${\\mathcal{Z}}$-stable.\nWe also don't know whether tracial ${\\mathcal{Z}}$-stability implies\nstable rank one, although this is expected to be true.\n\nThere is nothing special about the specific formulas\nfor $x \\mapsto \\alpha_x$\nin Example~\\ref{E_9214_ShiftAndUHF}\nand Example~\\ref{E_9Z26_ShiftAndUHF_v2},\nand $x \\mapsto \\beta_x$ in Example~\\ref{E_9214_ShiftAndCStFI_2}.\nThey were chosen merely to show that interesting examples exist.\n\nExample~\\ref{E_9214_ShiftAndUHF},\nExample~\\ref{E_9Z26_ShiftAndUHF_v2},\nand Example~\\ref{E_9214_ShiftAndCStFI_2}\nwere constructed so that the homeomorphism~$h$\ndoes not have mean dimension zero.\nIf $X$ is finite dimensional,\nthen one can get all we do by using known results\nfor crossed products by actions with finite Rokhlin dimension\nwith commuting towers.\nWith finite dimensional~$X$, one even gets ${\\mathcal{Z}}$-stability in\nexamples like Example~\\ref{E_9214_ShiftAndUHF},\nExample~\\ref{E_9Z26_ShiftAndUHF_v2},\nand Example~\\ref{E_9214_ShiftAndCStFI_2},\nby Theorem~5.8 of~\\cite{HWZ}.\nHowever,\nwe know of no results which apply to examples of this type\nwhen $X$ is infinite dimensional{} and $h$ has mean dimension zero.\n\nThe next most obvious choice for a minimal homeomorphism{} of an infinite dimensional{} space~$X$\nseems to be as follows.\nTake $X = (S^1)^{{\\mathbb{Z}}_{> 0}}$,\nfix $\\theta = (\\theta_n)_{n \\in {\\mathbb{Z}}_{> 0}} \\in {\\mathbb{R}}^{{\\mathbb{Z}}_{> 0}}$,\nand define $h \\colon X \\to X$ by\n$h \\bigl( (\\zeta_{n})_{n \\in {\\mathbb{Z}}_{> 0}} \\bigr)\n = \\bigl( e^{2 \\pi i \\theta_n} \\zeta_{n} \\bigr)_{n \\in {\\mathbb{Z}}_{> 0}}$.\nIf $1, \\theta_1, \\theta_2, \\ldots$ are linearly independent over~${\\mathbb{Q}}$,\nthen $h$ is minimal\nby Proposition~\\ref{P_4X04_ProdRotation}.\nHowever, by considering the action in just one coordinate,\none sees that, regardless of $\\zeta \\mapsto \\alpha_{\\zeta}$,\nthe action has finite Rokhlin dimension with commuting towers.\nBy Theorem 6.2 of~\\cite{HWZ}, irrational rotations\nhave finite Rokhlin dimension with commuting towers.\nRemark 6.3 of~\\cite{HWZ}, according to which the result\nextends to any homeomorphism which has an irrational\nrotation as a factor,\napplies equally well to any automorphism of $C (S^1) \\otimes B$,\nfor any unital C*-algebra~$B$,\nwhich lies over an irrational rotation on~$S^1$\nin the sense of Definition~\\ref{D_4Y12_OverX}.\nIf we start with $C \\bigl( (S^1)^{{\\mathbb{Z}}_{> 0}}, \\, D \\bigr)$,\ntake $B$ above to be $C \\bigl( (S^1)^{{\\mathbb{Z}}_{> 0} \\setminus \\{ 1 \\}}, \\, D \\bigr)$.\nTherefore Theorem~5.8 of~\\cite{HWZ} applies,\nand our results give nothing new.\n\nWe now give some examples of the second type discussed\nin the introduction to this section.\nFor easy reference,\nwe recall a result on the stable rank of reduced free products.\nMany reduced free products have stable rank~$1$.\nThe following result is from~\\cite{DHR}.\n(It is not affected by the correction~\\cite{DHRc}.)\nReduced free products of unital C*-algebra{s}\nin~\\cite{DHR}\nare implicitly taken to be amalgamated over~${\\mathbb{C}}$;\nsee Section~2.2 of~\\cite{DHR}.\n\n\\begin{prp}[Corollary~3.9 of~\\cite{DHR}]\\label{P_5416_TsrFrGp}\nLet $G$ and $H$ be discrete groups\nwith ${\\mathrm{card}} (G) \\geq 2$\nand ${\\mathrm{card}} (H) \\geq 3$.\nThen $C^*_{\\mathrm{r}} (G \\star H)$ has stable rank~$1$.\n\\end{prp}\n\n\\begin{exa}\\label{E_5416_CSrFGp}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nLet $\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\nbe as in Notation~\\ref{N_9214_FreeShift}.\nLet\n$\\alpha \\in {\\mathrm{Aut}} \\big( C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}) \\big)$\nbe the tensor product of the automorphism\n$f \\mapsto f \\circ h^{-1}$ of $C (X)$ and~$\\sigma$.\nThen\n$C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}),\n \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}),\n \\, \\alpha \\big)$\nhas stable rank~$1$.\n\nTo prove the claim,\nwe apply Theorem~\\ref{P_9Z26_XisCantor} with\n$D = C^*_{\\mathrm{r}} (F_{\\infty})$ and\n$\\alpha$ as given.\nUse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nBy Proposition 6.3.2 of \\cite{Rbt},\n$D$ has strict comparison of positive elements.\nBy construction, $\\alpha$ lies over~$h$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_StrCmp}).\n\\end{exa}\n\nApparently no previously known results give\nanything about the crossed product in this example.\nIn particular, as discussed after Problem~\\ref{Pb_9922_RokhlinTsr},\nknowing that the action has the Rokhlin property\ndoesn't seem to help.\n\nThere are many other automorphisms\nof $C^*_{\\mathrm{r}} (F_{\\infty})$\nwhich could be used in place of~$\\sigma$.\nFor example,\none could take a quasifree automorphism,\nas in Notation~\\ref{N_9214_QFreeFn}.\nHere is a more interesting version.\n\n\\begin{exa}\\label{E_5416_DenjFst}\nFix any $n \\in \\set{ 2, 3, \\ldots, \\infty }$\nand take $D = C^*_{\\mathrm{r}} (F_{n})$.\nAdopt Notation~\\ref{N_9214_QFreeFn}.\nLet $h \\colon X \\to X$ be a restricted Denjoy homeomorphism{}\nwith rotation number $\\theta \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nLet $\\zeta \\colon X \\to S^1$ be the continuous{} surjective map\nwith $\\zeta (h (x)) = e^{2 \\pi i \\theta} \\zeta (x)$\nof Lemma~\\ref{L_9X04_DjSemiCj}\n(gotten from Corollary~3.2 and Proposition~3.4 of~\\cite{PtSmdSk}).\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof~$D$,\nfor $x \\in X$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( D )$\nbe determined\nby $\\alpha_x (u_k) = \\zeta (x) u_k$\nfor $k = 1, 2, \\ldots, n$ (or $k \\in {\\mathbb{Z}}_{> 0}$ if $n = \\infty$).\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} \\big( C (X, D ) \\big)$,\nwhich we can think of as\na kind of noncommutative Furstenberg transformation.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X, D ), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, D ), \\, \\alpha \\big)$\nhas stable rank one.\nTo prove the claim,\nuse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\n\\end{exa}\n\nAgain, apparently no previously known results give\nanything about the crossed product.\nSince $C^*_{\\mathrm{r}} (F_{n})$ is not ${\\mathcal{Z}}$-stable,\nknowing that the action has the Rokhlin property\ndoesn't seem to help.\n\nWe can generalize Example~\\ref{E_5416_DenjFst} as follows.\n\n\\begin{exa}\\label{E_9217_MultltDenj}\nFix $n \\in \\{ 2, 3, \\ldots, \\infty \\}$.\nFix $\\theta_1, \\theta_2, \\ldots, \\theta_n \\in {\\mathbb{R}}$\n(or $\\theta_1, \\theta_2, \\ldots \\in {\\mathbb{R}}$)\nsuch that $1, \\theta_1, \\theta_2, \\ldots, \\theta_n$\n(or $1, \\theta_1, \\theta_2, \\ldots$)\nare linearly independent over~${\\mathbb{Q}}$.\nFor $k = 1, 2, \\ldots, n$,\nlet $h \\colon X_k \\to X_k$ be a restricted Denjoy homeomorphism{}\nwith rotation number $\\theta_k \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nTake $X = \\prod_{k = 1}^n X_k$,\nand let $h \\colon X \\to X$\nact as $h_k$ on the $k$-th factor.\nThen~$h$ is minimal by Proposition~\\ref{P_4X04_ProdRstDj}.\nDefine $\\zeta_k \\colon X_k \\to S^1$\nanalogously to the definition of $\\zeta$\nin Example~\\ref{E_5416_DenjFst},\nand take $\\alpha_x (u_k) = \\zeta_k (x_k) u_k$.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon\n {\\mathbb{Z}} \\to {\\mathrm{Aut}} \\big( C (X, \\, C^*_{\\mathrm{r}} (F_{n}) ) \\big)$.\nThen\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, \\, C^*_{\\mathrm{r}} (F_{n}) ),\n \\, \\alpha \\big)$\nis simple and has stable rank one\nfor the same reasons as in Example~\\ref{E_5416_DenjFst}.\n\\end{exa}\n\n\\begin{exa}\\label{R_5416_More}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nChoose any decomposition $X = X_1 \\amalg X_2$\nof $X$ as the disjoint union of two nonempty closed subsets.\nSet $D = C^*_{\\mathrm{r}} (F_{\\infty})$,\nwith the generators of $F_{\\infty}$ indexed by~${\\mathbb{Z}}$.\nFix any $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in (S^1)^{{\\mathbb{Z}}_{> 0}}$.\nFor $x \\in X_1$ take\n$\\alpha_{x}$ to be the automorphism\n$\\varphi_{\\zeta}$ of Notation~\\ref{N_9214_QFreeFn},\nexcept using ${\\mathbb{Z}}$ in place of ${\\mathbb{Z}}_{> 0}$,\nand for $x \\in X_2$ take\n$\\alpha_{x}$ to be the automorphism\n$\\sigma$ of Notation~\\ref{N_9214_FreeShift}.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} ( C (X, D) )$.\nThe algebra $C^* ({\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nThe algebra $D$ has strict comparison of positive elements by\nProposition 6.3.2 of~\\cite{Rbt},\nso $C^* ({\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha)$ has\nstable rank one\nby Theorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_StrCmp}).\n\\end{exa}\n\nExample~\\ref{R_5416_More} admits many variations.\nHere are several.\n\n\\begin{exa}\\label{E_9217_AltDenj}\nLet $h \\colon X \\to X$ be a restricted Denjoy homeomorphism,\nas in Example~\\ref{E_5416_DenjFst},\nand let $\\zeta \\colon X \\to S^1$ be as there.\nChoose any decomposition $X = X_1 \\amalg X_2$\nof $X$ as the disjoint union of two nonempty closed subsets.\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof $C^*_{\\mathrm{r}} (F_{2})$,\nfor $x \\in X_1$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe determined\nby $\\alpha_x (u_1) = \\zeta (x) u_1$ and $\\alpha_x (u_2) = u_2$,\nand for $x \\in X_2$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe determined\nby $\\alpha_x (u_1) = u_1$ and $\\alpha_x (u_2) = \\zeta (x) u_2$.\nThe algebra\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n \\otimes C^*_{\\mathrm{r}} (F_{2}), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\nthis algebra\nhas stable rank one.\nTo prove the claim,\nuse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\n\\end{exa}\n\n\\begin{exa}\\label{E_9217_DenjAndSw}\nIn Example~\\ref{E_9217_AltDenj},\nreplace the definition of $\\alpha_x$\nfor $x \\in X_2$ with\n$\\alpha_x (u_1) = u_2$ and $\\alpha_x (u_2) = u_1$.\nProposition~\\ref{P_4Y15_CPSimple}\nand Theorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen})\nstill apply,\nso\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n \\otimes C^*_{\\mathrm{r}} (F_{2}), \\, \\alpha \\big)$\nagain is simple and has stable rank one.\n\\end{exa}\n\nWe now give examples in which we also get real rank zero.\nThe following lemma will be used for them.\n\n\\begin{lem}\\label{L_0916_MakeSimple}\nLet $M$ be factor of type $\\mathrm{II}_1$,\nlet $\\alpha \\colon G \\to {\\mathrm{Aut}} (M)$\nbe an action of a countable group~$G$ on~$M$,\nand let $P \\S M$ be a separable C*-subalgebra.\nThen there exists a simple separable unital C*-subalgebra\n$D \\subset M$ which contains~$P$,\nis invariant under ${\\overline{\\sigma}}$,\nhas a unique tracial state\n(the restriction to $D$ of the unique tracial state on $M$),\nhas real rank zero and stable rank one,\nand such that the order on projections over~$D$\nis determined by traces.\n\\end{lem}\n\n\\begin{proof}\nThe proof is the same as that of Proposition~3.1 of~\\cite{Ph57},\nexcept for $G$-invariance.\nWe construct by induction on $n \\in {\\mathbb{Z}}_{\\geq 0}$ separable unital subalgebras\n$A_n, B_n, C_n, D_n, E_n, F_n \\S M$ with\n\\[\nP \\S A_0 \\S B_0 \\S C_0 \\S D_0 \\S E_0 \\S F_0 \\S \\cdots\n \\S A_n \\S B_n \\S C_n \\S D_n \\S E_n \\S F_n \\S \\cdots\n\\]\nsuch that $A_n$, $B_n$, $C_n$, $D_n$, and $E_n$ have the\nproperties in the proof of Proposition~3.1 of~\\cite{Ph57}\n($A_n$ is simple, etc.),\nand such that $F_n$ is $G$-invariant.\nIn the induction step, after constructing $E_n$,\nwe take $F_n$ to be the C*-subalgebra generated by\n$\\bigcup_{g \\in G} \\alpha_g (E_n)$.\nThen ${\\overline{\\bigcup_{n = 0}^{\\infty} F_n}}$ is $G$-invariant.\nThe proof in~\\cite{Ph57} now gives the desired conclusion,\nexcept that the conclusion is that $K_0 (D) \\to K_0 (M)$\nis an order isomorphism onto its range\ninstead of that the order on projections over~$D$\nis determined by traces.\nBut the conclusion we get implies the conclusion we want\nif $A$ has cancellation,\nand stable rank one implies cancellation\nby Proposition 6.4.1 and Proposition 6.5.1 of~\\cite{Blkd3}.\n\\end{proof}\n\n\\begin{exa}\\label{E_5416_DenjRR0}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nLet $\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\nbe as in Notation~\\ref{N_9214_FreeShift}.\n\nWe regard $C^*_{\\mathrm{r}} (F_{\\infty})$\nas a subalgebra of the group\nvon Neumann algebra $W^* (F_{\\infty})$\nin the usual way,\nand we let ${\\overline{\\sigma}} \\in {\\mathrm{Aut}} (W^* (F_{\\infty}))$\nbe the von Neumann algebra automorphism\nwhich shifts the generators of $F_{\\infty}$\nin the same way that $\\sigma$ does.\nThus ${\\overline{\\sigma}} |_{C^*_{\\mathrm{r}} (F_{\\infty})} = \\sigma$.\nChoose a ${\\overline{\\sigma}}$-invariant\nsubalgebra $D \\subset W^* (F_{\\infty})$\nwhich contains $C^*_{\\mathrm{r}} (F_{\\infty})$\nas in Lemma~\\ref{L_0916_MakeSimple},\nwith the properties there.\nSet $\\gamma = {\\overline{\\sigma}} |_D \\in {\\mathrm{Aut}} (D)$.\nThen define $\\alpha \\in {\\mathrm{Aut}} ( C (X) \\otimes D )$\nin the same was as in Example~\\ref{E_5416_CSrFGp},\nusing $\\gamma$ in place of~$\\sigma$.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nIt has stable rank~$1$ and real rank zero\nby Theorem \\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_SP}).\n\nThe algebra $D$ is not nuclear\nbecause the Gelfand-Naimark-Segal representation\nfrom its tracial state gives a nonhyperfinite factor.\n\\end{exa}\n\nIt is perhaps interesting to point out that every quasitrace\non the algebra~$D$ in Example~\\ref{E_5416_DenjRR0} is a trace.\nThis is a consequence of the following lemma.\n\n\\begin{lem}\\label{L_9929_UniqQTr}\nLet $A$ be a unital C*-algebra{} with real rank zero.\nSuppose that $\\tau$ is a quasitrace on~$A$ and\nthat whenever $p, q \\in M_{\\infty} (A)$ are projection{s}\nwith $\\tau (p) < \\tau (q)$,\nthen $p \\precsim q$.\nThen $\\tau$ is the only quasitrace on~$A$.\n\\end{lem}\n\n\\begin{proof}\nSuppose the conclusion is false,\nand let $\\sigma$ be some other quasitrace on~$A$.\nBy definition, for any quasitrace~$\\rho$ on~$A$\nand any $a, b \\in A_{\\mathrm{sa}}$,\nwe have $\\rho (a + i b) = \\rho (a) + i \\rho (b)$.\nTherefore there is $a \\in A_{\\mathrm{sa}}$\nsuch that $\\sigma (a) \\neq \\tau (a)$.\nSince quasitraces are continuous,\nit follows from real rank zero that there is $a \\in A_{\\mathrm{sa}}$\nsuch that $a$ has finite spectrum and $\\sigma (a) \\neq \\tau (a)$.\nSince quasitraces are linear on commutative C*-subalgebras,\nthere is a projection{} $p \\in A$ such that $\\sigma (p) \\neq \\tau (p)$.\n\nSuppose $\\sigma (p) < \\tau (p)$.\nChoose $m, n \\in {\\mathbb{Z}}_{\\geq 0}$ such that $m \\sigma (p) < n < m \\tau (p)$.\nDefine projection{s} $e, f \\in M_{\\infty} (A)$ by $e = 1_{M_m} \\otimes p$ and\n$f = 1_{M_n} \\otimes 1_A$.\nThen $\\tau (f) < \\tau (e)$, so $f \\precsim e$.\nBut $\\sigma (f) > \\sigma (e)$, a contradiction.\nSimilarly, $\\sigma (p) > \\tau (p)$ is also impossible.\nThis contradiction shows that $\\sigma$ does not exist.\n\\end{proof}\n\n\\begin{exa}\\label{E_9217_BiggerCstFn}\nLet $n \\in \\set{ 2, 3, \\ldots }$.\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof $C^*_{\\mathrm{r}} (F_{n})$,\nfor $\\rho$ in the symmetric group~$S_n$\nlet $\\psi_{\\rho} \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe the automorphism determined\nby $\\psi_{\\rho} (u_k) = u_{\\rho^{-1} (k)}$ for $k = 1, 2, \\ldots, n$,\nand let ${\\overline{\\psi_{\\rho} }}$ be the corresponding automorphism\nof the group von Neumann algebra $W^* (F_{n})$.\n\nBy Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes},\nthe algebra $C^*_{\\mathrm{r}} (F_{n})$ is simple\nand has a unique tracial state.\nUse Lemma~\\ref{L_0916_MakeSimple}\nto find a simple separable unital C*-algebra{}\n$D \\subset W^* (F_{n})$ which contains $C^*_{\\mathrm{r}} (F_{n})$,\nis invariant under all the automorphisms\n${\\overline{\\psi_{\\rho} }}$ for $\\rho \\in S_n$,\nand has the other properties given in Lemma~\\ref{L_0916_MakeSimple}.\n\nLet $X$ be the Cantor set, and let $h \\colon X \\to X$ be any minimal homeomorphism.\nChoose any decomposition $X = \\coprod_{\\rho \\in S_n} X_{\\rho}$\nof $X$ as the disjoint union of nonempty closed subsets.\nFor $x \\in X_{\\rho}$ let $\\alpha_x = {\\overline{\\psi_{\\rho} }} |_D$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\n\nThe algebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nIt has stable rank one and real rank zero by\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\nThe algebra $D$ is not nuclear\nfor the same reason as in Example~\\ref{E_5416_DenjRR0}.\n\\end{exa}\n\nThe next example is of the third type discussed\nin the introduction to this section.\n\n\\begin{exa}\\label{E_9217_GenPType}\nLet $(\\pi_1, \\pi_2, \\ldots)$\nbe a sequence of unital finite dimensional{} representations\nof $C^* (F_2)$\nsuch that, for every $n \\in {\\mathbb{Z}}_{> 0}$,\nthe representation\n$\\bigoplus_{k = n}^{\\infty} \\pi_k$ is faithful.\nFor $n \\in {\\mathbb{Z}}_{\\geq 0}$,\nlet $d (n)$ be the dimension of $\\pi_n$,\nand define\n$l (n) = d (n) + 4$\nand $r (n) = \\prod_{k = 1}^n l (k)$.\nLet $u_1, u_2 \\in C^* (F_2)$\nbe the ``standard'' unitaries in $C^* (F_2)$,\nobtained as the images of the standard generators of~$F_2$,\nas in Notation~\\ref{N_9214_FreeShift}\nexcept that we are now using the full C*-algebra{} instead\nof the reduced C*-algebra.\nLet $\\gamma_1, \\gamma_2, \\gamma_3 \\in {\\mathrm{Aut}} (C^* (F_2))$\nbe the automorphisms determined by\n\\[\n\\gamma_1 (u_1) = u_1^{-1}\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_1 (u_2) = u_2,\n\\]\n\\[\n\\gamma_2 (u_1) = u_1\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_2 (u_2) = u_2^{-1},\n\\]\nand\n\\[\n\\gamma_3 (u_1) = u_1^{-1}\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_3 (u_2) = u_2^{-1}.\n\\]\nFor $n \\in {\\mathbb{Z}}_{> 0}$\ndefine\n$D_n = M_{r (n)} \\otimes C^* (F_2)$,\nwhich we identify as\n\\[\nM_{l (n)} \\otimes M_{l (n - 1)} \\otimes \\cdots\n \\otimes M_{l (1)} \\otimes C^* (F_2),\n\\]\nand define\n$\\gamma_{n, n - 1} \\colon D_{n - 1} \\to D_n$\nby,\nfor $a \\in D_{n - 1} = M_{r (n - 1)} \\otimes C^* (F_2)$,\n\\begin{align*}\n\\gamma_{n, n - 1} (a)\n& = {\\mathrm{diag}} \\bigl( a, \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_1 ) (a),\n \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_2 ) (a),\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_3 ) (a),\n \\, (\\pi_n \\otimes {\\mathrm{id}}_{M_{r (n - 1)}} (a) \\otimes 1\n \\bigr)\n\\\\\n& \\in M_{l (n)} \\otimes D_{n - 1} = D_n.\n\\end{align*}\nLet $D$ be the direct limit of the resulting direct system.\nIt follows by methods of~\\cite{Ddlt}\nthat $D$ is simple, separable, and has tracial rank zero.\nIn particular, $D$ has stable rank one\nand real rank zero by Theorem 3.4 of~\\cite{HLinTrAF},\nand the order on projections over $D$ is determined by traces\nby Theorem 6.8 of~\\cite{HLinTTR}.\nMoreover, using the known result for $K_* (C^* (F_n))$\n(see~\\cite{CuFree}),\none gets $K_1 (D) = 0$.\n\nFor $\\zeta \\in S^1$ define inductively\nunitaries $u_n (\\zeta) \\in D_n$ as follows.\nSet $u_0 (\\zeta) = 1$,\nand, given $u_{n - 1} (\\zeta) \\in D_{n - 1}$,\nset\n\\[\nu_n (\\zeta)\n = \\bigl( {\\mathrm{diag}} \\bigl( \\zeta, 1, 1, \\ldots, 1 \\bigr)\n \\otimes 1_{D_{n - 1}} \\bigr)\n \\gamma_{n, n - 1} ( u_{n - 1} (\\zeta) )\n \\in M_{l (n)} \\otimes D_{n - 1} = D_n.\n\\]\nThen the actions\n$\\zeta \\mapsto {\\mathrm{Ad}} (u_n (\\zeta)) \\in {\\mathrm{Aut}} (D_n)$\nof $S^1$ on~$D_n$\nare compatible with the direct system,\nand so yield a continuous{} map\n(in fact, an action)\n$\\zeta \\mapsto \\alpha_{\\zeta} \\colon S^1 \\to {\\mathrm{Aut}} (D)$.\nLet $h \\colon S^1 \\to S^1$ be an irrational rotation.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} ( C (S^1, D) )$.\n\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (S^1, D), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}\nand has stable rank one\nby\nTheorem \\ref{T_9222_Tsr1AndRR0_CrPrd}(\\ref{9927_AyCentLargeConds_SP}).\n\\end{exa}\n\nMethods of~\\cite{NiuWng} will probably show that\nthe algebra $D$ in Example~\\ref{E_9217_GenPType} is not ${\\mathcal{Z}}$-stable\n(see Question~\\ref{Q_9217_GenPType_IsZStab}),\nalthough it definitely is tracially ${\\mathcal{Z}}$-stable.\n\nFinally, we give purely infinite examples.\n\n\\begin{exa}\\label{E_9214_ShiftAndOI}\nLet $(X, h)$ be the minimal subshift\n(of the shift on $(S^1)^{{\\mathbb{Z}}}$) with nonzero mean dimension\nused in Example~\\ref{E_9214_ShiftAndUHF}.\nLet $D = {\\mathcal{O}}_{\\infty}$,\nbut with standard generators indexed by~${\\mathbb{Z}}$,\nsay $s_k$ for $k \\in {\\mathbb{Z}}$.\nLet $\\alpha_{\\zeta}$ be the gauge automorphism,\ngiven by $\\alpha_{\\zeta} (s_k) = \\zeta_k s_k$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase}.\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis nuclear,\nit is necessarily ${\\mathcal{O}}_{\\infty}$-stable\nby Theorem 3.15 of \\cite{KiPh}.\n\\end{exa}\n\nIn Example~\\ref{E_9214_ShiftAndOI},\nTheorems 4.1 and~5.8 of~\\cite{HWZ} don't apply,\nsince there is no reason to think that $\\alpha$\nhas finite Rokhlin dimension with commuting towers.\n\n\\begin{exa}\\label{E_9214_OnPINotZStab}\nLet $D$ be the reduced free product\n$D = (M_2 \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$,\ntaken with respect to the Lebesgue measure state on $C ([0, 1])$\nand the state on $M_2 \\otimes M_2$\ngiven by tensor product of the usual tracial state ${\\mathrm{tr}}$\nwith the state\n$\\rho (x) = {\\operatorname{tr}} \\big(\n {\\mathrm{diag}} \\big( \\frac{1}{3}, \\frac{2}{3} \\big) x \\big)$\non $M_2$.\nIt is shown in Example 5.8 of~\\cite{AmGlJmPh}\nthat $D$ is purely infinite and simple\nbut not ${\\mathcal{Z}}$-stable.\n\nTake $X = (S^1)^4$,\nand for $\\zeta = (\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4) \\in X$\ntake $\\alpha_{\\zeta}$\nto be the free product automorphism\nwhich is given by\n\\[\n{\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n \\zeta_1 & 0 \\\\\n 0 & \\zeta_2\n \\end{matrix} \\right) \\right)\n\\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n \\zeta_3 & 0 \\\\\n 0 & \\zeta_4\n \\end{matrix} \\right) \\right)\n\\]\non $M_2 \\otimes M_2$\nand is trivial on $C ([0, 1])$.\nChoose $\\theta_1, \\theta_2, \\theta_3, \\theta_4 \\in {\\mathbb{R}}$\nsuch that $1, \\theta_1, \\theta_2, \\theta_3, \\theta_4$\nare linearly independent over~${\\mathbb{Q}}$.\nTake $h \\colon X \\to X$\nto be\n\\[\n(\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4)\n \\mapsto \\bigl( e^{2 \\pi i \\theta_1} \\zeta_1, \\,\n e^{2 \\pi i \\theta_2} \\zeta_2, \\,\n e^{2 \\pi i \\theta_3} \\zeta_3, \\,\n e^{2 \\pi i \\theta_4} \\zeta_4 \\bigr).\n\\]\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase}.\n\\end{exa}\n\nThe action in Example~\\ref{E_9214_OnPINotZStab}\nhas finite Rokhlin dimension with commuting towers.\nHowever, we don't know any theorem on pure infiniteness\nfor crossed products by such actions\nwhen the original algebra is not ${\\mathcal{O}}_{\\infty}$-stable.\nWe address this in Question~\\ref{Pb_0111_FinRDim_PI} below.\nOur result also does not imply that the crossed product\nis ${\\mathcal{O}}_{\\infty}$-stable, or even ${\\mathcal{Z}}$-stable,\nalthough it seems plausible that it might be.\n\nIn the next two examples, $D$ is again not ${\\mathcal{Z}}$-stable.\nAlso, $X$ isn't finite dimensional, and $h$ doesn't even have mean dimension zero.\nSo a positive answer to Question~\\ref{Pb_0111_FinRDim_PI}\npresumably would not help.\n\n\\begin{exa}\\label{E_9214_OnPINotZStab_2}\nLet $D$ be as in Example~\\ref{E_9214_OnPINotZStab},\nand let $(X, h)$ be as in Example~\\ref{E_9214_ShiftAndUHF}\n(and reused in Example~\\ref{E_9214_ShiftAndOI}).\nFor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X$\ntake $\\alpha_{\\zeta}$ to be the free product automorphism\nwhich is given by\n\\[\n{\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n \\zeta_1 & 0 \\\\\n 0 & \\zeta_2\n \\end{matrix} \\right) \\right)\n\\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n \\zeta_3 & 0 \\\\\n 0 & \\zeta_4\n \\end{matrix} \\right) \\right)\n\\]\non $M_2 \\otimes M_2$\nand is trivial on $C ([0, 1])$.\n(We are only using the coordinates with indexes $1, 2, 3, 4$.)\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase},\nbut may well not be ${\\mathcal{Z}}$-stable.\n\\end{exa}\n\nWith a small modification,\nwe can give an example of this type in which the\nunderlying action of $(S^1)^{{\\mathbb{Z}}}$\nis effective.\n\n\\begin{exa}\\label{E_9Z26_OnPINotZStab_v2}\nLet $B$ be the $2^{\\infty}$~UHF algebra,\nand let $\\tau$ be its (unique) tracial state.\nLet ${\\mathrm{tr}}$ be the tracial state{} on~$M_2$,\nand let $\\rho$ be the state on $M_2$ given by\n$\\rho (x)\n = {\\mathrm{tr}} \\big( {\\mathrm{diag}} \\big( \\frac{1}{3}, \\frac{2}{3} \\big) x \\big)$.\nLet $D$ be\nthe reduced free product\n$D = (B \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$,\ntaken with respect to the state $\\tau \\otimes \\rho$ on $B \\otimes M_2$\nand the Lebesgue measure state on $C ([0, 1])$.\nWe claim that $A$ is purely infinite and simple\nbut not ${\\mathcal{Z}}$-stable.\n\nTo prove pure infiniteness,\nin Examples 3.9(iii) of~\\cite{Dkm2}\ntake $A_1 = B$\nwith the state~$\\tau$,\ntake $F = M_2$\nwith the state~$\\rho$,\nand take $B = C ([0, 1])$\nwith the state given by Lebesgue measure.\nThese choices satisfy the hypotheses there.\nSo $A$ is is purely infinite and simple.\n\nTo prove that $A$ is not ${\\mathcal{Z}}$-stable,\nlet $\\pi$ be the Gelfand-Naimark-Segal representation of~$B$\nassociated with~$\\tau$,\nset $N = \\tau (B)''$,\nand also write $\\tau$ for the corresponding tracial state{} on~$N$.\nIn Proposition 5.6 of~\\cite{AmGlJmPh},\ntake $P_1 = N \\otimes M_2$\nwith the state $\\tau \\otimes \\rho$,\nand take $P_2 = L^{\\infty} ([0, 1])$\nwith the state given by Lebesgue measure.\nDefine\n\\[\na = \\left[ \\left( \\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix} \\right)\n \\otimes 1 \\otimes 1 \\otimes \\cdots \\right]\n \\otimes 1\n \\in \\left( \\bigotimes_{n = 1}^{\\infty} M_2 \\right) \\otimes M_2\n = B \\otimes M_2\n \\S P_1.\n\\]\nTake $G_1 = \\{ 1, a\\} \\S P_1$,\nand take $G_2 \\S P_2$ to be the set of functions\n$\\lambda \\mapsto e^{2 \\pi i n \\lambda}$ for $n \\in {\\mathbb{Z}}$.\nThese choices\nsatisfy the hypotheses there.\nMoreover, $G_1 \\S B \\otimes M_2$ and $G_2 \\S C ([0, 1])$.\nTherefore the reduced free product\n$A = (B \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$\nis a subalgebra of the algebra~$P$\nin Proposition 5.6 of~\\cite{AmGlJmPh}\nwhich contains $a$, $b$, and~$c$,\nso is not ${\\mathcal{Z}}$-stable by Proposition 5.6 of~\\cite{AmGlJmPh}.\nThe claim is proved.\n\nLet $(X, h)$ be\nas in Example~\\ref{E_9214_ShiftAndUHF}\n(and reused in Example~\\ref{E_9214_ShiftAndOI}).\nLet $\\sigma \\colon {\\mathbb{Z}}_{> 0} \\to {\\mathbb{Z}}$ be a bijection\n(as in Example~\\ref{E_9214_ShiftAndUHF}).\nFor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X$\ntake $\\alpha_{\\zeta}$\nto be the free product automorphism\nwhich is the identity on $C ([0, 1])$\nand which is the infinite tensor product\n\\[\n\\left[ \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}}\n {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n 1 & 0 \\\\\n 0 & \\zeta_{\\sigma (n)}\n \\end{matrix} \\right) \\right) \\right]\n \\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n \\zeta_{\\sigma (1)} & 0 \\\\\n 0 & \\zeta_{\\sigma (2)}\n \\end{matrix} \\right) \\right)\n \\in {\\mathrm{Aut}} (B \\otimes M_2)\n\\]\non~$B \\otimes M_2$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase},\nbut may well not be ${\\mathcal{Z}}$-stable.\n\\end{exa}\n\n\\section{Open problems}\\label{Sec_Qs}\n\nIn this section,\nwe collect some open questions\nsuggested by the examples and results in this paper.\n\n\\begin{pbm}\\label{Pb_9922_RokhlinTsr}\nLet $A$ be a unital C*-algebra,\nand let $\\alpha$ be an action of ${\\mathbb{Z}}$ on~$A$\nwhich has finite Rokhlin dimension with commuting towers.\nSuppose that $A$ has stable rank one and $C^* ({\\mathbb{Z}}, A, \\alpha)$ is simple.\nDoes it follow that $C^* ({\\mathbb{Z}}, A, \\alpha)$\nhas stable rank one?\n\\end{pbm}\n\nThis seems to be unknown even if $A$ is simple\n(in which case simplicity of $C^* ({\\mathbb{Z}}, A, \\alpha)$ is automatic)\nand $\\alpha$ has the Rokhlin property.\nWithout assuming simplicity of $C^* ({\\mathbb{Z}}, A, \\alpha)$,\nthe answer is definitely no.\nFor aperiodic homeomorphism{s} of the Cantor set\nwhose transformation group \\ca{s} don't have stable rank one,\nsee Theorem~3.1 of~\\cite{Pt1}\n(it is easy to construct examples there\nwhich have more than one minimal set)\nor Example~8.8 of~\\cite{Phl10}.\nCorollary~2.6 of~\\cite{Szb}\nimplies that the corresponding actions of~${\\mathbb{Z}}$\nhave Rokhlin dimension with commuting towers at most~$1$.\nIn fact, though,\nat least in Example~8.8 of~\\cite{Phl10} we get the Rokhlin property.\n\n\\begin{lem}\\label{L_9922_HasRokhlin}\nLet $h \\colon X \\to X$ be the aperiodic homeomorphism{}\nof the Cantor set in Example~8.8 of~\\cite{Phl10}.\nThen the induced automorphism of $C (X)$ has the Rokhlin property.\n\\end{lem}\n\n\\begin{proof}\nRecall from~\\cite{Phl10} that $X_1 = {\\mathbb{Z}} \\cup \\{\\pm \\infty \\}$,\n$h_1 \\colon X_1 \\to X_1$ is $n \\mapsto n + 1$,\n$X_2$ is the Cantor set,\n$h_2 \\colon X_2 \\to X_2$ is minimal,\n$X = X_1 \\times X_2$,\nand $h = h_1 \\times h_2$.\n\nLet $N \\in {\\mathbb{Z}}_{> 0}$.\nThe standard first return time construction provides\n\\[\nn, r (1), r (2), \\ldots, r (n) \\in {\\mathbb{Z}}_{> 0}\n\\]\nwith $N < r (1) < r (2) < \\cdots < r (n)$,\nand compact open subsets $Z_1, Z_2, \\ldots, Z_n \\S X_2$,\nsuch that\n\\[\nX_2 = \\coprod_{k = 1}^{n} \\coprod_{j = 0}^{r (k) - 1} h_{2}^j (Z_k).\n\\]\nThen the sets\n\\[\nX_1 \\times Z_1, \\quad X_1 \\times Z_2,\n \\quad \\ldots, \\quad X_1 \\times Z_n\n\\]\nform a system of Rokhlin towers for~$h$,\nwith heights $r (1), r (2), \\ldots, r (n)$,\nall of which exceed~$N$.\n\\end{proof}\n\nIn fact,\nit seems to be known that any aperiodic homeomorphism{}\nof the Cantor set~$X$\ninduces an automorphism of $C (X)$ with the Rokhlin property.\nWe have not found a reference,\nand we do not prove this here.\n\nThe following question asks for a plausible generalization\nof Lemma~\\ref{L_4Y11_Comp}.\n\n\\begin{qst}\\label{Qst1_20190103D}\nLet $D$ be a simple unital C*-algebra.\nLet $X$ be a compact metric space, let $G$ be a discrete group,\nand let $(g,x) \\mapsto gx$ be a minimal\nand essentially free action of $G$ on $X$.\nLet $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$ be an action of $G$\nwhich lies over the action of $G$ on $X$\nand which is pseudoperiodically generated.\nSet $A = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\n\nDoes it follow that for every $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nthere exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset A$\nsuch that $f \\precsim_A a$?\n\\end{qst}\n\nWe expect the techniques in the proof of Lemma \\ref{L_4Y11_Comp}\n(taking $Y$ to be the empty set) can be used to answer\nthis question in the affirmative.\n\n\\begin{qst}\\label{Q_9212_ExistsNotPseudoper}\nIs there a simple unital C*-algebra~$D$\nsuch that ${\\mathrm{Aut}} (D)$\nis not pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}?\n\\end{qst}\n\nPresumably such examples exist,\nbut we don't know of any.\nIndeed,\nwe don't see any reason why there should not be $\\alpha \\in {\\mathrm{Aut}} (D)$\nsuch that $\\set{\\alpha^n \\colon n \\in {\\mathbb{Z}}}$ is not pseudoperiodic.\n\n\\begin{qst}\\label{Q_9214_IsZStab}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$\nin Example~\\ref{E_9214_ShiftAndCStFI_2}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nWhat about crossed products by similarly constructed actions?\n\\end{qst}\n\n\\begin{qst}\\label{Q_9214_IsZStab_2}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n \\otimes C^*_{\\mathrm{r}} (F_{\\infty}), \\, \\alpha \\big)$\nin Example~\\ref{E_5416_CSrFGp}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nWhat about crossed products by similarly constructed actions?\nWhat about Example~\\ref{E_5416_DenjFst}?\n\\end{qst}\n\n\\begin{qst}\\label{Q_9217_GenPType_IsZStab}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (S^1, D), \\, \\alpha \\big)$\nin Example~\\ref{E_9217_GenPType}.\nIs this algebra ${\\mathcal{Z}}$-stable?\n\\end{qst}\n\nThe following question\nis motivated by Example~\\ref{E_9214_OnPINotZStab}.\n\n\\begin{qst}\\label{Pb_0111_FinRDim_PI}\nLet $A$ be a nonsimple unital C*-algebra{} which is purely infinite\nin the sense of Definition~4.1 of~\\cite{KiRo2000}.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (A)$\nbe an action with finite Rokhlin dimension with commuting towers.\nDoes it follow that $C^* ({\\mathbb{Z}}, A, \\alpha)$\nis purely infinite?\n\\end{qst}\n\nIf $A$ is ${\\mathcal{O}}_{\\infty}$-stable,\nthen $C^* ({\\mathbb{Z}}, A, \\alpha)$ is at least ${\\mathcal{Z}}$-stable\nby Theorem~5.8 of~\\cite{HWZ}.\nIf $A$ is also exact,\nthen so is $C^* ({\\mathbb{Z}}, A, \\alpha)$\n(by Proposition 7.1(v) of~\\cite{Krh2}),\nand $C^* ({\\mathbb{Z}}, A, \\alpha)$ is traceless\n(as at the beginning of Section~5 of~\\cite{Rdm7})\nbecause $A$ is,\nso $C^* ({\\mathbb{Z}}, A, \\alpha)$ is purely infinite\nby Corollary~5.1 of~\\cite{Rdm7}.\n(In this case, one obviously wants ${\\mathcal{O}}_{\\infty}$-stability.\nSee Question~\\ref{Pb_0111_PI_ZStab_to_OIStab} below.)\nBut the question as stated seems to be open,\neven if one assumes that $C^* ({\\mathbb{Z}}, A, \\alpha)$ is simple.\nIf $A$ itself is simple,\nthen $C^* ({\\mathbb{Z}}, A, \\alpha)$\nis purely infinite even just assuming that $\\alpha$ is pointwise outer,\nby Corollary~4.4 of~\\cite{JngOsk}.\nProvided one uses the reduced crossed product,\nthis remains true if ${\\mathbb{Z}}$ is replaced by any discrete group.\n\n\\begin{qst}\\label{Pb_0111_OIStabQ}\nConsider the crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin Example~\\ref{E_9214_OnPINotZStab_2}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nIs it ${\\mathcal{O}}_{\\infty}$-stable?\n\\end{qst}\n\nWe hope that ${\\mathcal{Z}}$-stability should come from the\naction in the ``$(S^1)^4$~direction''.\nWe suppose that if a purely infinite simple C*-algebra{}\nis ${\\mathcal{Z}}$-stable,\nthen it is probably ${\\mathcal{O}}_{\\infty}$-stable,\nbut this seems to be open in general.\n\n\\begin{qst}\\label{Pb_0111_PI_ZStab_to_OIStab}\nLet $B$ be a ${\\mathcal{Z}}$-stable purely infinite simple C*-algebra.\nDoes it follow that $B$ is ${\\mathcal{O}}_{\\infty}$-stable?\n\\end{qst}\n\nIf $B$ is separable and nuclear,\nthe answer is yes, by Theorem~5.2 of~\\cite{Rdm7}.\nBut this result doesn't help with Example~\\ref{E_9214_OnPINotZStab_2},\neven if we prove ${\\mathcal{Z}}$-stability there.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n \\label{sec:Introduction}\n\nThe analysis of gravitational instability in the expanding universe\nmodel was first presented by Lifshitz in 1946 in a general relativistic \ncontext \\cite{Lifshitz}.\nHistorically, the much simpler, and in hindsight, more intuitive \nNewtonian study followed later \\cite{Bonnor}.\nThe pioneering study by Lifshitz is based on a gauge choice which is\ncommonly called the synchronous gauge. \nAs the later studies have shown, the synchronous gauge is only \none way of fixing the gauge freedom out of several available\nfundamental gauge conditions\n\\cite{Harrison,Nariai,Bardeen-1980,KS,Reviews,PRW}.\nAs will be summarized in the following, out of the several gauge conditions\nonly the synchronous gauge fails to fix the gauge mode completely and \nthus often needs more involved algebra.\nAs long as one is careful of the algebra this gauge choice does \nnot cause any kind of intrinsic problem; there exist, however, some persisting\nalgebraic errors in the literature, see Sec. \\ref{sec:Discussion}.\nThe gauge condition which turns out to be especially suitable for\nhandling the perturbed density is the comoving gauge used first by\nNariai in 1969 \\cite{Nariai}.\nSince the comoving gauge condition completely fixes the gauge transformation\nproperty, the variables in this gauge can be equivalently considered as\ngauge-invariant ones.\nAs mentioned, there exist several such fundamental gauge conditions\neach of which completely fixes the gauge transformation properties.\nOne of such gauge conditions suitable for handling the gravitational\npotential and the velocity perturbations is the zero-shear gauge\nused first by Harrison in 1967 \\cite{Harrison}.\nThe variables in such gauge conditions are equivalently gauge-invariant.\nUsing the gauge freedom as an advantage for handling the problem\nwas emphasized by Bardeen in 1988 \\cite{Bardeen-1988}.\nIn order to use the gauge choice as an advantage a {\\it gauge ready method}\nwas proposed in \\cite{PRW} which will be adopted in the \nfollowing, see Sec. \\ref{sec:Relativistic}.\n\nThe variables which characterize the self gravitating Newtonian fluid flow are\nthe density, the velocity and the gravitational potential \n(the pressure is specified by an equation of state),\nwhereas, the relativistic flow may be characterized by various components\nof the metric (and consequent curvatures) and the energy-momentum tensor.\nSince the relativistic gravity theory is a constrained system we have the \nfreedom of imposing certain conditions on the metric or the energy-momentum \ntensor as coordinate conditions. \nIn the perturbation analyses the freedom arises because we need to introduce\na fictitious background system in order to describe the physical perturbed\nsystem.\nThe correspondence of a given spacetime point between the perturbed spacetime\nand the ficticious background one could be made with certain degrees \nof freedom.\nThis freedom can be fixed by suitable conditions (the gauge conditions)\nbased on the spacetime coordinate transformation.\nStudies in the literature show that a certain variable in a certain gauge\ncondition correctly reproduces the corresponding Newtonian behavior.\nAlthough the perturbed density in the comoving gauge shows the Newtonian\nbehavior, the perturbed potential and the perturbed velocity in the\nsame gauge do not behave like the Newtonian ones; for example, in the comoving\ngauge the perturbed velocity vanishes by the coordinate (gauge) condition.\nIt is known that both the perturbed potential and the perturbed velocity\nin the zero-shear gauge correctly behave like the corresponding\nNewtonian variables \\cite{Bardeen-1980}.\n\nIn the first part (Sec. \\ref{sec:Six-gauges}) we will elaborate establishing \nsuch correspondences between the relativistic and Newtonian perturbed \nvariables.\nOur previous work on this subject is presented in \\cite{MDE,Hyun};\ncompared with \\cite{MDE} in this work we will explicitly compare \nthe relativistic equations for the perturbed density, potential and velocity \nvariables in several available gauge conditions with the\nones in the Newtonian system.\nWe will include both the background spatial curvature ($K$) \nand the cosmological constant ($\\Lambda$).\nIn the second part (Sec. \\ref{sec:Hydrodynamics}), using the variables \nwith correct Newtonian correspondences,\nwe will extend our relativistic results to the situations\nwith general pressures in the background and perturbations.\nWe will present the relativistic equations satisfied by the gauge-invariant\nvariables and will derive the {\\it general solutions} valid in the \nsuper-sound-horizon scale (i.e., larger than Jeans scale)\nconsidering both $K$ and $\\Lambda$, and the generally evolving\nideal fluid equation of state $p = p(\\mu)$.\n\nSection \\ref{sec:review} is a review of our previous work\ndisplaying the basic equations in both Newtonian and relativistic contexts\nand summarizing our strategy for handling the equations. \nIn Sec. \\ref{sec:Six-gauges} we consider a pressureless limit of the\nrelativistic equations.\nWe derive the equations for the perturbed density, the perturbed potential \nand the perturbed velocity in several different fundamental gauge conditions.\nBy comparing these relativistic equations in several gauges with\nthe Newtonian ones we {\\it identify} the gauge conditions which reproduce \nthe correct Newtonian behavior for certain variables in general scales.\nIn Sec. \\ref{sec:Hydrodynamics} we present the fully relativistic equations \nfor the identified gauge-invariant variables with correct Newtonian limits, \nnow, considering the general pressures in the background and perturbations.\nWe derive the general large-scale solutions valid for general $K$ and $\\Lambda$\nin an ideal fluid medium, which are thus valid for general equation of \nstate of the \nform $p = p(\\mu)$, but with negligible entropic and anisotropic pressures.\nThe solutions are valid in the super-sound-horizon scale, thus are valid \neffectively in all scales in the matter dominated era where the\nsound-horizon (Jeans scale) is negligible.\nWe discuss several quantities which are conserved in the large-scale\nunder the changing background equation of state.\nThese are the perturbed three-space curvature in several gauge conditions,\nand in particular, we find the three-space curvature perturbation\nin the comoving gauge shows a distinguished behavior:\nfor $K = 0$ (but for general $\\Lambda$), it is conserved in a\nsuper-sound-horizon scale independently of the changing equation of state.\n{}For completeness we also summarize the case with multiple fluid\ncomponents in Sec. \\ref{sec:Multi}, and cases of the rotation and\nthe gravitational wave in Sec. \\ref{sec:rot-GW}.\nSec. \\ref{sec:Hydrodynamics} is the highlight of the present work.\nSec. \\ref{sec:Discussion} is a discussion.\nWe set $c \\equiv 1$.\n\n\n\\section{Basic Equations and Strategy} \n \\label{sec:review}\n\n\n\\subsection{Newtonian Cosmological Perturbations} \n \\label{sec:Newtonian}\n\nSince the Newtonian perturbation analysis in the expanding medium is well known\nwe begin by summarizing the basic equations \\cite{MDE,Weinberg,Peebles-1980}.\nThe background evolution is governed by\n\\begin{eqnarray}\n & & H^2 = {8 \\pi G \\over 3} \\varrho - {K \\over a^2} + {\\Lambda \\over 3}, \n \\quad \\varrho \\propto a^{-3},\n \\label{Newt-BG}\n\\end{eqnarray}\nwhere we allowed the general curvature (total energy) \nand the cosmological constant; $a(t)$ is a cosmic scale factor,\n$H (t) \\equiv \\dot a \/ a$, and $\\varrho (t)$ is the mass density. \nIn Newtonian theory the cosmological constant can be introduced in the\nPoisson equation {\\it by hand} as \n$\\nabla^2 \\Phi = 4 \\pi G \\varrho - \\Lambda$.\nPerturbed parts of the mass conservation, the momentum conservation\nand the Poisson's equations are [see Eqs. (43,46) in \\cite{MDE}]: \n\\begin{eqnarray}\n & & \\delta \\dot \\varrho + 3 H \\delta \\varrho = - {k \\over a} \\varrho\n \\delta v, \\quad\n \\delta \\dot v + H \\delta v = {k \\over a} \\delta \\Phi, \\quad\n - {k^2 \\over a^2} \\delta \\Phi = 4 \\pi G \\delta \\varrho,\n \\label{Newt-eqs} \n\\end{eqnarray}\nwhere $\\delta \\varrho ({\\bf k}, t)$, $\\delta v ({\\bf k}, t)$ and\n$\\delta \\Phi ({\\bf k},t)$ are the Fourier modes of the perturbed mass density, \nvelocity, and gravitational potential, respectively \n[see Eqs. (44,45) in \\cite{MDE}];\n$k$ is a comoving wave number with $\\nabla^2 \\rightarrow - k^2\/a^2$.\n[For linear perturbations the same forms of equations are valid in the \nconfiguration and the Fourier spaces.\nThus, without causing any confusion, we often ignore distinguishing the\nFourier space from the configuration space by an additional subindex.]\nEquation (\\ref{Newt-eqs}) can be arranged into the closed form equations\nfor $\\delta$ ($\\equiv \\delta \\varrho \/ \\varrho$), $\\delta v$ and\n$\\delta \\Phi$ as:\n\\begin{eqnarray}\n & & \\ddot \\delta + 2 H \\dot \\delta - 4 \\pi G \\varrho \\delta \n = {1 \\over a^2 H} \\left[ a^2 H^2 \\left( {\\delta \\over H} \\right)^\\cdot\n \\right]^\\cdot = 0,\n \\label{Newt-delta-eq} \\\\\n & & \\delta \\ddot v + 3 H \\delta \\dot v + \\left( \\dot H + 2 H^2 \n - 4 \\pi G \\varrho \\right) \\delta v = 0,\n \\label{Newt-v-eq} \\\\\n & & \\delta \\ddot \\Phi + 4 H \\delta \\dot \\Phi\n + \\left( \\dot H + 3 H^2 - 4 \\pi G \\varrho \\right) \\delta \\Phi \n = {1 \\over a^3 H} \\left[ a^2 H^2 \n \\left( {a \\over H} \\delta \\Phi \\right)^\\cdot \\right]^\\cdot = 0.\n \\label{Newt-Phi-eq} \n\\end{eqnarray}\nWe note that Eqs. (\\ref{Newt-eqs}-\\ref{Newt-Phi-eq}) are\nvalid for general $K$ and $\\Lambda$.\nAlthough redundant, we presented these equations for later comparison\nwith the relativistic results.\nThe general solutions for $\\delta$, $\\delta v$ and $\\delta \\Phi$\nimmediately follow as (see also Table 1 in \\cite{MDE}):\n\\begin{eqnarray}\n & & \\delta ({\\bf k},t) = k^2 \\left[ \n H C ({\\bf k}) \\int^t_0 {dt \\over \\dot a^2} \n + {H \\over 4 \\pi G \\varrho a^3} d ({\\bf k}) \\right], \n \\label{delta-N} \\\\\n & & \\delta v ({\\bf k}, t) = - \\left[ {k \\over aH} C ({\\bf k})\n \\left( 1 + a^2 H \\dot H \\int^t_0 {dt \\over \\dot a^2} \\right)\n + {k \\dot H \\over 4 \\pi G \\varrho a^2} d ({\\bf k}) \\right],\n \\label{v-N} \\\\\n & & \\delta \\Phi ({\\bf k}, t) = - \\left[ 4 \\pi G \\varrho a^2 H C ({\\bf k}) \n \\int^t_0 {dt \\over \\dot a^2} + {H \\over a} d ({\\bf k}) \\right].\n \\label{Phi-N}\n\\end{eqnarray}\nThe coefficients $C ({\\bf k})$ and $d ({\\bf k})$ are two integration constants, \nand indicate the relatively growing and decaying solutions, respectively;\nthe coefficients are matched in accordance with the solutions with a general\npressure in Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) using \nEq. (\\ref{Correspondences-2})\n(different dummy variables for $t$ in the integrands are assumed).\nIn the literature we can find various analytic form expressions of the \nabove solutions in diverse situations, \nsee \\cite{Lifshitz,Harrison,Nariai,Weinberg,Peebles-1980,MDE-sols}.\n\n\n\\subsection{Relativistic Cosmological Perturbations} \n \\label{sec:Relativistic}\n\nIn the relativistic theory the fundamental variables are the metric and\nthe energy-momentum tensor.\nAs a way of summarizing our notation we present our convention of\nthe metric and the energy-momentum tensor.\nAs the metric we consider a spatially homogeneous and isotropic spacetime\nwith the most general perturbation\n\\begin{eqnarray}\n d s^2 = - a^2 \\left( 1 + 2 \\alpha \\right) d \\eta^2\n - a^2 \\left( \\beta_{,\\alpha} +B_\\alpha \\right) d \\eta d x^\\alpha\n + a^2 \\left[ g_{\\alpha\\beta}^{(3)} \\left( 1 + 2 \\varphi \\right)\n + 2 \\gamma_{|\\alpha\\beta} + 2 C_{(\\alpha|\\beta)} + 2 C_{\\alpha\\beta}\n \\right] d x^\\alpha d x^\\beta,\n \\label{metric} \n\\end{eqnarray}\nwhere $0 = \\eta$ with $dt \\equiv a d \\eta$,\nIndices $\\alpha$, $\\beta$ $\\dots$ are spatial ones and\n$g^{(3)}_{\\alpha\\beta}$ is the three-space metric of the homogeneous and\nisotropic space.\n$\\alpha ({\\bf x}, t)$, $\\beta ({\\bf x}, t)$, $\\gamma ({\\bf x}, t)$,\nand $\\varphi ({\\bf x}, t)$ indicate the scalar-type structure with\nfour degrees of freedom.\nThe transverse $B_\\alpha ({\\bf x}, t)$ and $C_\\alpha ({\\bf x}, t)$\nindicate the vector-type structure with four degrees of freedom.\nThe transverse-tracefree $C_{\\alpha\\beta} ({\\bf x}, t)$\nindicates the tensor-type structure with two degrees of freedom.\nThe energy-momentum tensor is decomposed as:\n\\begin{eqnarray}\n & & T^0_0 \\equiv - \\bar \\mu - \\delta \\mu, \\quad\n T^0_\\alpha \\equiv - {1 \\over k} \n \\left( \\bar \\mu + \\bar p \\right) v_{,\\alpha}\n + \\delta T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha},\n \\nonumber \\\\\n & & T^\\alpha_\\beta\n \\equiv \\left( \\bar p + \\delta p \\right) \\delta^\\alpha_\\beta \n + {1 \\over a^2} \\left( \\nabla^\\alpha \\nabla_\\beta\n - {1 \\over 3} \\Delta \\delta^\\alpha_\\beta \\right) \\sigma\n + \\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n + \\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta},\n \\label{Tab}\n\\end{eqnarray}\nwhere $\\mu ({\\bf k}, t) \\equiv \\bar \\mu (t) + \\delta \\mu ({\\bf k}, t)$ and \n$p \\equiv \\bar p + \\delta p$ are the energy density and the pressure, \nrespectively; an overbar indicates a background order quantity and will \nbe ignored unless necessary.\n$v$ is a frame-independent velocity related perturbed order variable\nand $\\sigma$ is an anisotropic pressure.\nSituation with multiple fluids will be considered in Sec. \\ref{sec:Multi}.\n$\\delta T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha}$, $\\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}$,\nand $\\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta}$ are vector and tensor-type\nperturbed order energy-momentum tensor.\nAll the spatial tensor indices and the vertical bar are based on \n$g^{(3)}_{\\alpha\\beta}$ as the metric.\nThe three types of structures are related to the density condensation,\nthe rotation, and the gravitational wave, respectively.\nDue to the high symmetry in the background the three types of perturbations\ndecouple from each other to the linear order.\nSince these three types of structures evolve independently, we can handle\nthem separately.\nEvolutions of the rotation and the gravitational wave will be discussed\nin Sec. \\ref{sec:rot-GW}.\n\n{}For the background equations we have [Eq. (6) in \\cite{Ideal}]:\n\\begin{eqnarray}\n H^2 = {8 \\pi G \\over 3} \\mu - {K \\over a^2} + {\\Lambda \\over 3}, \\quad\n \\dot H = - 4 \\pi G ( \\mu + p ) + {K \\over a^2}, \\quad\n \\dot \\mu = - 3 H \\left( \\mu + p \\right). \n \\label{BG-eqs} \n\\end{eqnarray}\nThese equations follow from $G^0_0$ and $G^\\alpha_\\alpha - 3 G^0_0$\ncomponents of Einstein equations and $T^b_{0;b} = 0$, respectively;\nthe third equation follows from the first two equations.\n{}For $p = 0$ and replacing $\\mu$ with $\\varrho$ Eq. (\\ref{BG-eqs}) \nreduces to Eq. (\\ref{Newt-BG}).\nThe relativistic cosmological perturbation equations without fixing\nthe temporal gauge condition, thus in a gauge ready form,\nwere derived in Eq. (22-28) of \\cite{PRW}\n[see also Eqs. (8-14) in \\cite{Ideal}; we use \n$\\delta \\equiv \\delta \\mu \/\\mu$ and $v \\equiv - (k\/a) \\Psi\/(\\mu + p)$]:\n\\begin{eqnarray}\n & & \\kappa \\equiv 3 \\left( - \\dot \\varphi + H \\alpha \\right)\n + {k^2 \\over a^2} \\chi,\n \\label{eq1} \\\\\n & & - {k^2 - 3K \\over a^2} \\varphi + H \\kappa = - 4 \\pi G \\mu \\delta,\n \\label{eq2} \\\\\n & & \\kappa - {k^2 - 3K \\over a^2} \\chi = 12 \\pi G \\left( \\mu + p \\right)\n {a \\over k} v,\n \\label{eq3} \\\\\n & & \\dot \\chi + H \\chi - \\alpha - \\varphi = 8 \\pi G \\sigma,\n \\label{eq4} \\\\\n & & \\dot \\kappa + 2 H \\kappa = \\left( {k^2 \\over a^2} - 3 \\dot H \\right) \n \\alpha + 4 \\pi G \\left( 1 + 3 c_s^2 \\right) \\mu \\delta \n + 12 \\pi G e,\n \\label{eq5} \\\\\n & & \\dot \\delta + 3 H \\left( c_s^2 - {\\rm w} \\right) \\delta \n + 3 H { e \\over \\mu } = \\left( 1 + {\\rm w} \\right)\n \\left( \\kappa - 3 H \\alpha - {k \\over a} v \\right),\n \\label{eq6} \\\\\n & & \\dot v + \\left( 1 - 3 c_s^2 \\right) H v = {k \\over a} \\alpha\n + {k \\over a \\left( 1 + {\\rm w} \\right) } \\left( c_s^2 \\delta\n + {e \\over \\mu} - {2\\over 3} {k^2 - 3K \\over a^2} {\\sigma \\over \\mu}\n \\right),\n \\label{eq7} \n\\end{eqnarray}\nwhere we introduced a spatially gauge-invariant combination\n$\\chi \\equiv a ( \\beta + a \\dot \\gamma)$.\nAccording to their origins, Eqs. (\\ref{eq1}-\\ref{eq7}) can be called: \nthe definition of $\\kappa$, \nADM energy constraint ($G^0_0$ component of Einstein equations), \nmomentum constraint ($G^0_\\alpha$ component), \nADM propagation \n($G^\\alpha_\\beta - {1 \\over 3} \\delta^\\alpha_\\beta G^\\gamma_\\gamma$ component),\nRaychaudhuri ($G^\\alpha_\\alpha - G^0_0$ component), \nenergy conservation ($T^b_{0;b} = 0$),\nand momentum conservation ($T^b_{\\alpha;b} = 0$) equations, respectively.\nThe perturbed metric variables $\\varphi ({\\bf k}, t)$, $\\kappa ({\\bf k},t)$,\n$\\chi ({\\bf k}, t)$ and $\\alpha ({\\bf k}, t)$ are the perturbed part of\nthe three space curvature, expansion, shear, and the lapse function, \nrespectively; meanings for $\\varphi$, $\\kappa$, and $\\chi$ are based on\nthe normal frame vector field.\nThe isotropic pressure is decomposed as\n\\begin{eqnarray}\n \\delta p ({\\bf k}, t) \\equiv c_s^2 (t) \\delta \\mu ({\\bf k}, t) \n + e ({\\bf k}, t), \\quad\n c_s^2 \\equiv {\\dot p \\over \\dot \\mu}, \\quad\n {\\rm w} (t) \\equiv {p \\over \\mu}.\n \\label{p-decomposition}\n\\end{eqnarray}\n\nUnder the gauge transformation $\\tilde x^a = x^a + \\xi^a$, the variables\ntransform as (see Sec. 2.2 in \\cite{PRW}):\n\\begin{eqnarray}\n & & \\tilde \\alpha = \\alpha - \\dot \\xi^t, \\quad\n \\tilde \\varphi = \\varphi - H \\xi^t, \\quad\n \\tilde \\chi = \\chi - \\xi^t, \\quad\n \\tilde \\kappa = \\kappa \n + \\left( 3 \\dot H - {k^2 \\over a^2} \\right) \\xi^t,\n \\nonumber \\\\\n & & \\tilde v = v - {k \\over a} \\xi^t, \\quad\n \\tilde \\delta = \\delta + 3 \\left( 1 + {\\rm w} \\right) H \\xi^t,\n \\label{GT}\n\\end{eqnarray}\nwhere $\\xi^t \\equiv e^s \\xi^\\eta$, and $e$ and $\\sigma$ are \ngauge-invariant.\n[Due to the spatial homogeneity of the background, the effect \n from the spatial gauge transformation has been trivially handled; \n $\\chi$ and $v$ are the spatially gauge-invariant combinations of \n the variables, and the other metric and fluid variables \n are naturally spatially gauge-invariant \\cite{Bardeen-1988}.]\nThus, the perturbed variables in Eqs. (\\ref{eq1}-\\ref{eq7})\nare {\\it designed} so that any one of the following conditions\ncan be used to fix the freedom based on the temporal gauge transformation:\n$\\alpha \\equiv 0$ (the synchronous gauge), \n$\\varphi \\equiv 0$ (the uniform-curvature gauge),\n$\\chi \\equiv 0$ (the zero-shear gauge), \n$\\kappa \\equiv 0$ (the uniform-expansion gauge), \n$v\/k \\equiv 0$ (the comoving gauge), \nand $\\delta \\equiv 0$ (the uniform-density gauge).\nThese gauge conditions contain most of the gauge conditions used in the\nliterature concerning the cosmological hydrodynamic perturbation in a single\ncomponent situation; for the multiple component case see Eq. (\\ref{Gauges-2}).\nOf course, we can make an infinite number of different linear combinations \nof these gauge conditions each of which is also suitable for the temporal \ngauge fixing condition.\nOur reason for choosing these conditions as fundamental is partly based \non conventional use, but apparently, as the names of the conditions indicate,\nthe gauge conditions also have reasonable meanings based on the metric and \nthe energy-momentum tensors.\n\nEach one of these gauge conditions, except for the synchronous gauge, \nfixes the temporal gauge transformation property completely.\nThus, each variable in these five gauge conditions \nis equivalent to a corresponding gauge-invariant combination\n(of the variable concerned and the variable used in the gauge condition).\nWe proposed a convenient way of writing the gauge-invariant variables \n\\cite{PRW}.\n{}For example, we let\n\\begin{eqnarray}\n & & \\delta_v \\equiv \\delta + 3 (1 + {\\rm w}) {aH \\over k} v\n \\equiv 3 (1 + {\\rm w}) {a H \\over k} v_\\delta, \\quad\n \\varphi_\\chi \\equiv \\varphi - H \\chi \\equiv - H \\chi_\\varphi, \\quad\n v_\\chi \\equiv v - {k \\over a} \\chi \\equiv - {k \\over a} \\chi_v.\n \\label{GI-combinations}\n\\end{eqnarray}\nThe variables $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ are gauge-invariant \ncombinations; $\\delta_v$ becomes $\\delta$ in the comoving gauge which takes\n$v\/k=0$ as the gauge condition, etc.\nIn this manner we can systematically construct the corresponding \ngauge-invariant combination for any variable based on a gauge condition which \nfixes the temporal gauge transformation property completely.\nOne variable evaluated in different gauges can be considered as different \nvariables, and they show different behaviors in general.\nIn this sense, except for the synchronous gauge, the variables in the \nrest of five gauges can be considered as the gauge-invariant variables.\n[The variables with a subindex $\\alpha$ are not gauge-invariant, \nbecause those are equivalent to variables in the synchronous gauge.]\nAlthough $\\delta_v$ is a gauge-invariant variable which becomes\n$\\delta$ in the comoving gauge, $\\delta_v$ itself \ncan be evaluated in any gauge with the same value.\n\nEquations (\\ref{eq1}-\\ref{eq7}) are the basic set of gauge-ready form\nequations for hydrodynamic perturbations proposed in \\cite{PRW}: \n``The moral is that one should work in the gauge that is mathematically\nmost convenient for the problem at hand'', \\cite{Bardeen-1988}.\nAccordingly, in order to use the available gauge conditions as the\nadvantage, Eqs. (\\ref{eq1}-\\ref{eq7}) are designed \nfor easy implementation of the fundamental gauge conditions.\nUsing this {\\it gauge-ready formulation}, complete sets of solutions for the \nsix different gauge conditions are presented in an ideal fluid case \n\\cite{Ideal}, and in a pressureless medium \\cite{MDE}.\nEquations (\\ref{BG-eqs}-\\ref{p-decomposition}) include general pressures which \nmay account for the nonequilibrium or dissipative effects in hydrodynamic \nflows in the cosmological context with general $K$ and $\\Lambda$.\nThe equations are expressed in general forms so that the fluid quantities \ncan represent successfully the effects of the scalar field and \nclasses of generalized gravity theories \\cite{PRW}.\nThe applications of the gauge-ready formalism to the minimally coupled scalar\nfield and to the generalized gravity theories have been made in \\cite{MSF-GGT}.\n\n\n\\section{Newtonian Correspondence of the Relativistic Analyses} \n \\label{sec:Six-gauges}\n\nIn \\cite{MDE,Hyun} we developed arguments on the correspondences\nbetween the Newtonian and the relativistic analyses.\nIn order to reinforce the Newtonian correspondences of certain\n(gauge-invariant) variables in certain gauges, in the following we\nwill present the closed form differential equations for \n$\\delta$, $v$ and $\\varphi$ in the six different gauge conditions\nand compare them with the Newtonian ones in \nEqs. (\\ref{Newt-delta-eq}-\\ref{Newt-Phi-eq}).\n\nThere are reasons why we should know about possible Newtonian behaviors\nof the relativistic variables.\nVariables in the relativitistic gravity are only parameters appearing in\nthe metric and energy-momentum tensor.\n{}For example, later we will find that only in the zero-shear gauge does\nthe perturbed curvature variable behave as the perturbed Newtonian potential,\nwhich we have some experience of in Newtonian physics.\nThe same variable evaluated in other gauge conditions is simply other\nvariable and we cannot regard it as the perturbed potential.\nAs we have introduced several different fundamental gauge conditions\nit became necessary to identify the correct gauge where a variable shows\nthe corresponding Newtonian behavior.\nAs will be shown in this section, we rarely find Newtonian correspondences.\nIn fact, there exists an almost unique gauge condition for each\nrelativistic perturbation variable which shows Newtonian behavior.\nThe results will be summarized in Sec. \\ref{sec:Correspondences}.\n\nIn this section we consider a {\\it pressureless medium},\n$p = 0$ (thus ${\\rm w} = 0 = c_s^2$) and $e = 0 = \\sigma$.\nHowever, we consider general $K$ and $\\Lambda$.\n\n\n\\subsection{Comoving Gauge} \n \\label{sec:CG}\n\nAs the gauge condition we set $v\/k \\equiv 0$.\nEquivalently, we can set $v\/k = 0$ and leave the other variables as\nthe gauge-invariant combinations with subindices $v\/k$ or simply $v$.\n{}From Eq. (\\ref{eq7}) we have $\\alpha_v = 0$.\nThus the comoving gauge is a case of the synchronous gauge; this is\ntrue {\\it only} in a pressureless situation.\n{}From Eqs. (\\ref{eq1}-\\ref{eq7}) we can derive:\n\\begin{eqnarray}\n & & \\ddot \\delta_v + 2 H \\dot \\delta_v - 4 \\pi G \\mu \\delta_v = 0,\n \\label{CG-delta-eq} \\\\\n & & \\ddot \\varphi_v + 3 H \\dot \\varphi_v - {K \\over a^2} \\varphi_v = 0.\n \\label{CG-varphi-eq} \n\\end{eqnarray}\nThus, Eq. (\\ref{CG-delta-eq}) has a form identical to Eq. \n(\\ref{Newt-delta-eq}).\nHowever, Eq. (\\ref{CG-varphi-eq}) differs from Eq. (\\ref{Newt-Phi-eq}),\nand apparently we do not have an equation for $v$.\n\n{}For $K = 0$ Eq. (\\ref{CG-varphi-eq}) leads to\ntwo solutions which are $\\varphi_v \\propto {\\rm constant}$ and \n$\\int_0^t a^{-3} dt$. \nSince we have $\\alpha_v = 0$, for $K = 0$, from Eqs. (\\ref{eq1},\\ref{eq3})\nwe have $\\dot \\varphi_v = 0$.\nThis implies that, for $K = 0$, the second solution, \n$\\varphi_v \\propto \\int_0^t a^{-3} dt$, should have the vanishing coefficient,\nsee Eq. (\\ref{varphi_v-sol3}).\nThe solution of Eq. (\\ref{CG-varphi-eq}) for general $K$ will be\npresented later in Eq. (\\ref{varphi_v-sol}).\n\n\n\\subsection{Synchronous Gauge} \n \\label{sec:SG}\n\nWe let $\\alpha \\equiv 0$.\nThis gauge condition does not fix the temporal gauge transformation property\ncompletely.\nThus, although we can still indicate the variables in this gauge condition\nusing subindices $\\alpha$ without ambiguity, these variables are not \ngauge-invariant; see the discussion after Eq. (\\ref{varphi-GI}).\nEquation (\\ref{eq7}) leads to\n\\begin{eqnarray}\n v_\\alpha = c_g {k \\over a},\n\\end{eqnarray}\nwhich is a pure gauge mode.\nThus, fixing $c_g \\equiv 0$ exactly corresponds to the comoving gauge.\nWe can show that the following two equations are not affected by the\nremaining gauge mode.\n\\begin{eqnarray}\n & & \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha \n - 4 \\pi G \\mu \\delta_\\alpha = 0,\n \\label{SG-delta-eq} \\\\\n & & \\ddot \\varphi_\\alpha + 3 H \\dot \\varphi_\\alpha \n - {K \\over a^2} \\varphi_\\alpha = 0.\n \\label{SG-varphi-eq} \n\\end{eqnarray}\nEquation (\\ref{SG-delta-eq}) is identical to Eq. (\\ref{Newt-delta-eq}).\nThis is because in a pressureless medium the behavior of the gauge modes \nhappen to coincide with the behavior of the decaying physical solutions\nfor $\\delta_\\alpha$ and $\\varphi_\\alpha$.\nHowever, for the variables $\\kappa_\\alpha$ and $\\chi_\\alpha$ the gauge mode \ncontribution appears explicitly.\n\nThus, in a pressureless medium, variables\nin the synchronous gauge behave the same as the ones in the comoving gauge, \nexcept for the additional gauge modes which appear in the synchronous gauge\nfor some variables.\nIn a pressureless medium, we can simultaneously impose both the comoving gauge\nand the synchronous gauge conditions.\nHowever, this is possible only in a pressureless medium, see the Appendix.\nIn Sec. \\ref{sec:Discussion} we indicate some common errors in the literature\nbased on the synchronous gauge.\n\n\n\\subsection{Zero-shear Gauge} \n \\label{sec:ZSG}\n\nWe let $\\chi \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\chi$.\nWe can derive:\n\\begin{eqnarray}\n & & \\ddot \\delta_\\chi + { 2 k^2 \/ a^2 - 36 \\pi G \\mu \\left[ 1 + ( H^2 \n - 2 \\dot H ) a^2 \/ \\left( k^2 - 3 K \\right) \\right] \\over\n k^2 \/ a^2 - 12 \\pi G \\mu \\left[ 1 + 3 H^2 a^2\/\\left( k^2 - 3K \\right) \n \\right] }\n H \\left( \\dot \\delta_\\chi \n - 12 \\pi G \\mu {a^2 \\over k^2 - 3 K} H \\delta_\\chi \\right)\n \\nonumber \\\\\n & & \\qquad \n - 4 \\pi G \\mu \\left[ 1 - 3 ( H^2 - 2 \\dot H )\n {a^2 \\over k^2 - 3 K} \\right] \\delta_\\chi = 0,\n \\label{ZSG-delta-eq} \\\\\n & & \\ddot v_\\chi + 3 H \\dot v_\\chi \n + \\left( \\dot H + 2 H^2 - 4 \\pi G \\mu \\right) v_\\chi = 0,\n \\label{ZSG-v-eq} \\\\\n & & \\ddot \\varphi_\\chi + 4 H \\dot \\varphi_\\chi + \\left( \\dot H + 3 H^2 \n - 4 \\pi G \\mu \\right) \\varphi_\\chi = 0.\n \\label{ZSG-varphi-eq} \n\\end{eqnarray}\nThus, Eqs. (\\ref{ZSG-v-eq},\\ref{ZSG-varphi-eq}) have forms identical \nto Eqs. (\\ref{Newt-v-eq},\\ref{Newt-Phi-eq}), respectively.\nHowever, only in the small-scale limit is the behavior of $\\delta_\\chi$\nthe same as the Newtonian one.\n\n\n\\subsection{Uniform-expansion Gauge} \n \\label{sec:UEG}\n\nWe let $\\kappa \\equiv 0$, and substitute the remaining variables into\nthe gauge-invariant combinations with subindices $\\kappa$.\nWe can derive:\n\\begin{eqnarray}\n & & \\ddot \\delta_\\kappa \n + \\left( 2 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n \\right) \\dot \\delta_\\kappa - \\left[ 4 \\pi G \\mu {k^2\/a^2 + 6 H^2 \\over\n k^2\/a^2 - 3 \\dot H} + \\left( {12 \\pi G \\mu H \\over\n k^2\/a^2 - 3 \\dot H} \\right)^\\cdot \\right] \\delta_\\kappa = 0,\n \\label{UEG-delta-eq} \\\\ \n & & \\ddot v_\\kappa + \\left[ 2 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n + {4 \\pi G \\mu \\over k^2\/a^2 - 3 \\dot H}\n \\left( {k^2\/a^2 - 3 \\dot H \\over 4 \\pi G \\mu} \\right)^\\cdot\n \\right] \\dot v_\\kappa\n \\nonumber \\\\ \n & & \\quad \n + \\left[ \\dot H + H^2 - 4 \\pi G \\mu {k^2\/a^2 + 3 H^2 \\over\n k^2\/a^2 - 3 \\dot H} + H {4 \\pi G \\mu \\over k^2\/a^2 - 3 \\dot H}\n \\left( { k^2\/a^2 - 3 \\dot H \\over 4 \\pi G \\mu} \\right)^\\cdot\n \\right] v_\\kappa = 0,\n \\label{UEG-v-eq} \\\\ \n & & \\ddot \\varphi_\\kappa \n + \\left( 4 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n \\right) \\dot \\varphi_\\kappa\n + \\left[ \\dot H + 3 H^2 - 4 \\pi G \\mu {k^2\/a^2 + 9 H^2 \\over\n k^2\/a^2 - 3 \\dot H} - \\left( {12 \\pi G \\mu H \\over k^2\/a^2\n - 3 \\dot H} \\right)^\\cdot \\right] \\varphi_\\kappa = 0.\n \\label{UEG-varphi-eq}\n\\end{eqnarray}\nIn the small-scale limit we can show that\nEqs. (\\ref{UEG-delta-eq}-\\ref{UEG-varphi-eq}) \nreduce to Eqs. (\\ref{Newt-delta-eq}-\\ref{Newt-Phi-eq}), respectively.\nThus, {\\it in the small-scale limit}, all three variables $\\delta_\\kappa$, \n$v_\\kappa$ and $\\varphi_\\kappa$ correctly reproduce the Newtonian behavior.\nHowever, outside or near the (visual) horizon scale, the behaviors of all these\nvariables {\\it strongly deviate} from the Newtonian ones.\n\nIn Sec. 84 of \\cite{Peebles-1980} we find that in order to get the usual \nNewtonian equations a coordinate transformation is made so that\nwe have $\\dot h \\equiv 0$ in the new coordinate.\nWe can show that $\\dot h = 2 \\kappa$ in our notation.\nThus, the new coordinate used in \\cite{Peebles-1980} is the uniform-expansion \ngauge\\footnote{This was incorrectly pointed out after Eq. (49) in \\cite{MDE}:\n In \\cite{MDE} it was wishfully mentioned that in Sec. 84 of \n \\cite{Peebles-1980} the gauge transformations were made into the \n comoving gauge for $\\delta$ and into the zero-shear gauge for \n $v$ and $\\varphi$, respectively.\n }.\n\n\n\\subsection{Uniform-curvature Gauge} \n \\label{sec:UCG}\n\nWe let $\\varphi \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\varphi$.\nWe have\n\\begin{eqnarray}\n & & \\ddot \\delta_\\varphi + 2 H { (k^2 - 3K)\/a^2 + 18 \\pi G \\mu \\over \n (k^2 - 3K)\/a^2 + 12 \\pi G \\mu } \\dot \\delta_\\varphi\n - { 4 \\pi G \\mu k^2\/a^2 \\over \n (k^2 - 3K)\/a^2 + 12 \\pi G \\mu } \\delta_\\varphi = 0,\n \\label{UCG-delta-eq} \\\\\n & & \\ddot v_\\varphi + \\left( 5 H + 2 {\\dot H \\over H} \\right) \\dot v_\\varphi\n + \\left( 3 \\dot H + 4 H^2 \\right) v_\\varphi = 0. \n \\label{UCG-v-eq}\n\\end{eqnarray}\nIn the small-scale limit Eq. (\\ref{UCG-delta-eq}) reduces to\nEq. (\\ref{Newt-delta-eq}).\nIn the uniform-curvature gauge, the perturbed potential is set to be equal\nto zero by the gauge condition.\nThe uniform-curvature gauge condition is known to have \ndistinct properties in handling the scalar field or the dilaton field \nwhich appears in a broad class of the generalized gravity theories\n\\cite{MSF-GGT,GRG-MSF-GGT}.\n\n\n\\subsection{Uniform-density Gauge} \n \\label{sec:UDG}\n\nWe let $\\delta \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\delta$.\nWe have\n\\begin{eqnarray}\n & & \\ddot v_\\delta + 2 \\left( 2 H + {\\dot H \\over H} \\right) \\dot v_\\delta\n + 3 \\left( H^2 + \\dot H \\right) v_\\delta = 0,\n \\label{UDG-v-eq} \\\\\n & & \\ddot \\varphi_\\delta + 2 H { (k^2 - 3 K)\/a^2 + 18 \\pi G \\mu\n \\over (k^2- 3K)\/a^2 + 12 \\pi G \\mu} \\dot \\varphi_\\delta\n - {4 \\pi G \\mu k^2\/a^2\n \\over (k^2- 3K)\/a^2 + 12 \\pi G \\mu} \\varphi_\\delta = 0.\n \\label{UDG-varphi-eq} \n\\end{eqnarray}\nThese equations differ from Eqs. (\\ref{Newt-v-eq},\\ref{Newt-Phi-eq}).\nOf course, we have no equation for $\\delta$ which is set to be equal to zero \nby our choice of the gauge condition.\n\n\n\\subsection{Newtonian Correspondences} \n \\label{sec:Correspondences}\n\nAfter a thorough comparison made in this section we found that \nequations for $\\delta$ in the comoving gauge ($\\delta_v$), and\nfor $v$ and $\\varphi$ in the zero-shear gauge ($v_\\chi$ and $\\varphi_\\chi$)\nshow the same forms as the corresponding Newtonian equations \n{\\it in general scales} and considering general $K$ and $\\Lambda$.\nUsing these gauge-invariant combinations in Eq. (\\ref{GI-combinations}),\nEqs. (\\ref{eq1}-\\ref{eq7}) can be combined to give:\n\\begin{eqnarray}\n \\dot \\delta_v = - {k^2 - 3K \\over a k} v_\\chi, \\quad\n \\dot v_\\chi + H v_\\chi = - {k \\over a} \\varphi_\\chi, \\quad\n {k^2 - 3K \\over a^2} \\varphi_\\chi = 4 \\pi G \\mu \\delta_v.\n \\label{GI-eqs}\n\\end{eqnarray}\nComparing Eq. (\\ref{GI-eqs}) with Eq. (\\ref{Newt-eqs}) \nwe can identify either one of the following correspondences:\n\\begin{eqnarray}\n & & \\delta_v \\leftrightarrow \\delta |_{\\rm Newtonian}, \\quad\n {k^2 - 3 K \\over k^2} v_\\chi \n \\leftrightarrow \\delta v |_{\\rm Newtonian}, \\quad\n - {k^2 - 3 K \\over k^2} \\varphi_\\chi \\leftrightarrow \n \\delta \\Phi |_{\\rm Newtonian},\n \\label{Correspondences-1} \\\\\n & & {k^2 \\over k^2 - 3 K} \\delta_v \\leftrightarrow \n \\delta |_{\\rm Newtonian}, \\quad\n v_\\chi \\leftrightarrow \\delta v |_{\\rm Newtonian}, \\quad\n - \\varphi_\\chi \\leftrightarrow \\delta \\Phi |_{\\rm Newtonian}.\n \\label{Correspondences-2}\n\\end{eqnarray}\nIn \\cite{MDE} we proposed the correspondences in Eq. (\\ref{Correspondences-1}),\nbut the ones in Eq. (\\ref{Correspondences-2}) also work well.\nIn fact, the gravitational potential identified in \nEq. (\\ref{Correspondences-2}) is the one often found in the Newtonian limit \nof the general relativity, e.g. in \\cite{Weinberg}.\nUsing Eqs. (\\ref{GI-combinations},\\ref{eq3},\\ref{eq4},\\ref{GI-eqs}) \nwe can also identify the following relations:\n\\begin{eqnarray}\n & & \\delta_v = 3 H {a \\over k} v_\\delta, \\quad\n v_\\chi = - {k \\over a} \\chi_v = - {ak \\over k^2 - 3 K} \\kappa_v, \\quad\n \\varphi_\\chi = -\\alpha_\\chi = - H \\chi_\\varphi.\n\\end{eqnarray}\n\n{}From a given solution known in one gauge we can derive all the rest of the \nsolutions even in other gauge conditions.\nThis can be done either by using the gauge-invariant combination of variables \nor directly through gauge transformations.\nThe complete set of exact solutions in a pressureless medium is presented \nin \\cite{MDE}.\n{}From our study in this section and using the complete solutions presented \nin Tables 1 and 2 of \\cite{MDE} we can identify variables in certain gauges \nwhich correspond to the Newtonian ones. These are summarized in Table 1.\n\n\\centerline{ [[{\\bf TABLE 1.}]] }\n\n\\noindent\nAs mentioned earlier, all three variables in the uniform-expansion gauge show \nNewtonian correspondence in the small-scale; however all of them change \nthe behaviors near and above the horizon scale. \nIn the small-scale limit, except for the \nuniform-density gauge where $\\delta \\equiv 0$, \n$\\delta$ in all gauge conditions behaves in the same way \\cite{Bardeen-1980}.\nMeanwhile, since only $\\delta_v$ ($\\delta$ in the comoving gauge)\nshows the Newtonian behavior in a general scale, it may be natural to identify \n$\\delta_v$ as the one most closely resembling the Newtonian one.\n\nNotice that, although we have horizons in the relativistic analysis\nthe equations for $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$\nkeep the same form as the corresponding Newtonian equations\n{\\it in the general scale}.\nConsidering this as an additional point we regard\n$\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ as the one most closely corresponding\nto the Newtonian variables.\nThis argument will become stronger as we consider the case with a general\npressure in the next section.\n\n\n\\section{Relativistic Cosmological Hydrodynamics}\n \\label{sec:Hydrodynamics}\n\n\n\\subsection{General Equations and Large-scale Solutions}\n \\label{sec:Eqs}\n\nIn the previous sections we have shown that the gauge-invariant\ncombinations $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ behave most similarly\nto the Newtonian $\\delta \\equiv \\delta \\varrho\/\\varrho$, \n$\\delta v$ and $\\delta \\Phi$, respectively.\nThe equations remain the same in a general scale which includes the\nsuperhorizon scales in the relativistic situation considering \ngeneral $K$ and $\\Lambda$.\nIn this section, we will present the fully general relativistic equations \nfor $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ including the effects of the \ngeneral pressure terms.\nEquations (\\ref{eq1}-\\ref{eq7}) are the basic set of \nperturbation equations in a gauge-ready form.\n\n{}From Eqs. (\\ref{eq3},\\ref{eq6},\\ref{eq7}),\nEqs. (\\ref{eq4},\\ref{eq7}), Eqs. (\\ref{eq2},\\ref{eq3}), and\nEqs. (\\ref{eq1},\\ref{eq3},\\ref{eq4}), we have, respectively:\n\\begin{eqnarray}\n & & \\dot \\delta_v - 3 H {\\rm w} \\delta_v\n = - \\left( 1 + {\\rm w} \\right) {k \\over a} {k^2 - 3 K \\over k^2} v_\\chi\n - 2 H {k^2 - 3K \\over a^2} {\\sigma \\over \\mu},\n \\label{P-1} \\\\\n & & \\dot v_\\chi + H v_\\chi = - {k \\over a} \\varphi_\\chi\n + {k \\over a \\left( 1 + {\\rm w} \\right)} \n \\left[ c_s^2 \\delta_v + {e \\over \\mu}\n - 8 \\pi G \\left( 1 + {\\rm w} \\right) \\sigma\n - {2 \\over 3} {k^2 - 3 K \\over a^2} {\\sigma \\over \\mu} \\right],\n \\label{P-2} \\\\\n & & {k^2 - 3 K \\over a^2} \\varphi_\\chi = 4 \\pi G \\mu \\delta_v,\n \\label{P-3} \\\\\n & & \\dot \\varphi_\\chi + H \\varphi_\\chi\n = - 4 \\pi G \\left( \\mu + p \\right) {a \\over k} v_\\chi \n - 8 \\pi G H \\sigma.\n \\label{P-4}\n\\end{eqnarray}\nConsidering either one of the correspondences in \nEqs. (\\ref{Correspondences-1},\\ref{Correspondences-2})\nwe immediately see that Eqs. (\\ref{P-1}-\\ref{P-4}) have the \ncorrect Newtonian limit expressed in Eqs. (\\ref{Newt-eqs},\\ref{GI-eqs}).\nEquations (\\ref{P-1}-\\ref{P-4}) were presented in \\cite{Bardeen-1980}.\n\nCombining Eqs. (\\ref{P-1}-\\ref{P-4}) we can derive closed\nform expressions for the $\\delta_v$ and $\\varphi_\\chi$\nwhich are the relativistic counterpart of Eqs. \n(\\ref{Newt-delta-eq},\\ref{Newt-Phi-eq}).\nWe have:\n\\begin{eqnarray}\n & & \\ddot \\delta_v + \\left( 2 + 3 c_s^2 - 6 {\\rm w} \\right) H \\dot \\delta_v\n + \\Bigg[ c_s^2 {k^2 \\over a^2} \n - 4 \\pi G \\mu \\left( 1 - 6 c_s^2 + 8 {\\rm w} - 3 {\\rm w}^2 \\right)\n + 12 \\left( {\\rm w} - c_s^2 \\right) {K \\over a^2}\n + \\left( 3 c_s^2 - 5 {\\rm w} \\right) \\Lambda \\Bigg] \\delta_v\n \\nonumber \\\\\n & & \\qquad \\qquad\n = {1 + {\\rm w} \\over a^2 H} \\left[ {H^2 \\over a (\\mu + p)} \n \\left( {a^3 \\mu \\over H} \\delta_v \\right)^\\cdot \\right]^\\cdot\n + c_s^2 {k^2 \\over a^2} \\delta_v\n \\nonumber \\\\\n & & \\qquad \\qquad\n = - {k^2 - 3 K \\over a^2} { 1 \\over \\mu }\n \\left\\{ e + 2 H \\dot \\sigma\n + 2 \\left[ - {1 \\over 3} {k^2 \\over a^2} + 2 \\dot H\n + 3 \\left( 1 + c_s^2 \\right) H^2 \\right] \\sigma \\right\\},\n \\label{delta-eq}\\\\\n & & \\ddot \\varphi_\\chi + \\left( 4 + 3 c_s^2 \\right) H \\dot \\varphi_\\chi\n + \\left[ c_s^2 {k^2 \\over a^2} \n + 8 \\pi G \\mu \\left( c_s^2 - {\\rm w} \\right)\n - 2 \\left( 1 + 3 c_s^2 \\right) {K \\over a^2}\n + \\left( 1 + c_s^2 \\right) \\Lambda \\right] \\varphi_\\chi\n \\nonumber \\\\\n & & \\qquad \\qquad\n = {\\mu + p \\over H} \\left[ {H^2 \\over a (\\mu + p)} \n \\left( {a \\over H} \\varphi_\\chi \\right)^\\cdot \\right]^\\cdot\n + c_s^2 {k^2 \\over a^2} \\varphi_\\chi\n = {\\rm stresses}.\n \\label{varphi-eq}\n\\end{eqnarray}\nThese two equations are related by Eq. (\\ref{P-3}) which resembles the \nPoisson equation.\nNotice the remarkably compact forms presented in the second steps of\nEqs. (\\ref{delta-eq},\\ref{varphi-eq}).\nIt may be an interesting exercise to show that the above equations\nare indeed valid for general $K$ and $\\Lambda$, and\nfor the general equation of state $p = p(\\mu)$;\nuse $\\dot {\\rm w} = - 3 H (1 + {\\rm w}) ( c_s^2 - {\\rm w} )$.\nEquation (\\ref{delta-eq}) became widely known through Bardeen's seminal paper \nin \\cite{Bardeen-1980}.\nIn a less general context but originally \nEqs. (\\ref{delta-eq},\\ref{varphi-eq}) were derived by\nNariai \\cite{Nariai} and Harrison \\cite{Harrison}, respectively;\nhowever, the compact expressions are new results in this paper.\n\nIf we ignore the entropic and anisotropic pressures ($e = 0 = \\sigma$) \non scales larger than Jeans scale\nEq. (\\ref{varphi-eq}) immediately leads to a general integral form \nsolution as\n\\begin{eqnarray}\n \\varphi_\\chi ({\\bf k}, t)\n &=& 4 \\pi G C ({\\bf k}) {H \\over a} \\int_0^t {a (\\mu + p) \\over H^2} dt\n + {H \\over a} d ({\\bf k})\n \\nonumber \\\\\n &=& C ({\\bf k}) \\left[ 1 - {H \\over a} \\int_0^t\n a \\left( 1 - {K \\over \\dot a^2} \\right) dt \\right]\n + {H \\over a} d ({\\bf k}).\n \\label{varphi_chi-sol}\n\\end{eqnarray}\nThis solution was first derived in Eq. (108) of \n\\cite{HV}\\footnote{We correct a typographical \n error in Eq. (129) of \\cite{HV}: $c_0\/a$ should be replaced by $c_0$.\n },\nsee also Eq. (55) in \\cite{PRW}.\nSolutions for $\\delta_v$ and $v_\\chi$ follow from Eqs.\n(\\ref{P-3},\\ref{P-4}), respectively, as\n\\begin{eqnarray}\n & & \\delta_v ({\\bf k}, t) = {k^2 - 3 K \\over 4 \\pi G \\mu a^2} \n \\varphi_\\chi ({\\bf k}, t), \n \\label{delta_v-sol} \\\\\n & & v_\\chi ({\\bf k}, t)= - {k \\over 4 \\pi G ( \\mu + p) a^2}\n \\left\\{ C ({\\bf k}) \\left[ {K \\over \\dot a} \n - \\dot H \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt \\right]\n + \\dot H d ({\\bf k}) \\right\\}.\n \\label{v_chi-sol}\n\\end{eqnarray}\nWe stress that these large-scale asymptotic solutions are\n{\\it valid for the general $K$, $\\Lambda$, and $p = p(\\mu)$};\n$C({\\bf k})$ and $d({\\bf k})$ are integration constants considering \nthe general evolution of $p = p(\\mu)$.\nWe also emphasize that {\\it the large-scale criterion is the sound-horizon},\nand thus in the matter dominated era the solutions are valid even far \ninside the (visual) horizon as long as the scales are larger than Jeans scale.\nIn the pressureless limit, considering Eq. (\\ref{Correspondences-2}),\nEqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol})\ncorrectly reproduce Eqs. (\\ref{delta-N}-\\ref{Phi-N}).\n\nUsing the set of gauge-ready form equations in\nEqs. (\\ref{eq1}-\\ref{eq7}) the complete set of corresponding general\nsolutions for all variables in all different gauges can be easily derived;\nfor such sets of solutions with less general assumptions see \\cite{MDE,Ideal}.\n{}For $K = 0 = \\Lambda$ and ${\\rm w} = {\\rm constant}$ we have\n$a \\propto t^{2\/[3(1+{\\rm w})]}$ and Eqs. \n(\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) become:\n\\begin{eqnarray}\n \\varphi_\\chi ({\\bf k}, t) \n &=& {3 (1 + {\\rm w}) \\over 5 + 3 {\\rm w}} C ({\\bf k})\n + {2 \\over 3 (1 + {\\rm w})} {1 \\over at} d ({\\bf k})\n \\nonumber \\\\\n &\\propto& {\\rm constant}, \n \\quad t^{- {5 + 3 {\\rm w} \\over 3 ( 1 + {\\rm w} )} }\n \\quad \\propto \\quad {\\rm constant}, \n \\quad a^{- {5 + 3 {\\rm w} \\over 2} },\n \\nonumber \\\\\n \\delta_v ({\\bf k}, t) \n &\\propto& t^{2(1 + 3 {\\rm w}) \\over 3 (1 + {\\rm w})}, \n \\quad t^{- {1 - {\\rm w} \\over 1 + {\\rm w} } }\n \\quad \\propto \\quad a^{1 + 3 {\\rm w}}, \n \\quad a^{- {3 \\over 2} (1 - {\\rm w}) },\n \\nonumber \\\\\n v_\\chi ({\\bf k}, t)\n &\\propto& t^{1 + 3 {\\rm w} \\over 3 (1 + {\\rm w})}, \n \\quad t^{- {4 \\over 3(1 + {\\rm w}) } }\n \\quad \\propto \\quad a^{1 + 3 {\\rm w} \\over 2}, \n \\quad a^{- 2}.\n \\label{Sol-w=const}\n\\end{eqnarray}\nThese solutions were presented in \\cite{Bardeen-1980} and\nin less general contexts but originally in \\cite{Harrison,Nariai}.\n{}For ${\\rm w} =0$ we recover the well known Newtonian behaviors.\n \n\n\\subsection{Conserved Quantities}\n \\label{sec:Conservation}\n\nIn an ideal fluid, the curvature fluctuations in several gauge conditions \nare known to be conserved in the {\\it superhorizon scale} independently\nof the changes in the background equation of state.\n{}From Eqs. (41,73) in \\cite{Ideal} we find [for $K = 0 = \\Lambda$,\nbut for general $p(\\mu)$]:\n\\begin{eqnarray}\n \\varphi_v = \\varphi_\\delta = \\varphi_\\kappa = \\varphi_\\alpha = C ({\\bf k}),\n \\label{varphi-conserv}\n\\end{eqnarray}\nand the dominating decaying solutions vanish (or, are cancelled).\nThe combinations follow from Eq. (\\ref{GT}) as:\n\\begin{eqnarray}\n \\varphi_v = \\varphi - {aH \\over k} v, \\quad\n \\varphi_\\delta \\equiv \\varphi +{\\delta \\over 3 (1 + {\\rm w})} \n \\equiv \\zeta, \\quad\n \\varphi_\\kappa \\equiv \\varphi + {H \\over 3 \\dot H - k^2\/a^2 } \\kappa, \n \\quad\n \\varphi_\\alpha \\equiv \\varphi - H \\int^t \\alpha dt.\n \\label{varphi-GI}\n\\end{eqnarray}\nThe $\\varphi_\\alpha$ combination which is $\\varphi$ in the synchronous gauge\nis not gauge-invariant; the lower bound of integration gives the behavior\nof the gauge mode; in Eq. (\\ref{varphi-conserv}) we ignored the gauge mode.\nThe large-scale conserved quantity, $\\zeta$, proposed in\n\\cite{Bardeen-1988,BST} is $\\varphi_\\delta$.\nIn \\cite{Ideal} the above conservation properties are shown assuming \n$K = 0 = \\Lambda$; in such a case the integral form solutions for a complete\nset of variables are presented in Table 8 in \\cite{Ideal}.\nIn Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) we have the large-scale \nintegral form solutions in the case of general $K$ and $\\Lambda$.\nThus, now we can see the behavior of these variables considering the \ngeneral $K$ and $\\Lambda$.\n\n{}Evaluating $\\varphi_v$ in Eq. (\\ref{varphi-GI}) in the zero-shear gauge,\nthus $\\varphi_v = \\varphi_\\chi - (aH \/ k) v_\\chi$, and using\nthe solutions in Eqs. (\\ref{varphi_chi-sol},\\ref{v_chi-sol}) we \ncan derive\n\\begin{eqnarray}\n \\varphi_v ({\\bf k}, t) \n &=& C ({\\bf k}) \\left\\{ 1 + {K \\over a^2}\n {1 \\over 4 \\pi G (\\mu + p)} \\left[ 1 \n - {H \\over a} \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt \n \\right] \\right\\}\n + {K \\over a^2} { H\/a \\over 4 \\pi G ( \\mu + p)} d ({\\bf k})\n \\nonumber \\\\\n &=& C ({\\bf k}) \\left[ 1 + {K \\over a^2}\n {H\/a \\over \\mu + p} \\int_0^t {a (\\mu + p) \\over H^2} dt \\right]\n + {K \\over a^2} { H\/a \\over 4 \\pi G ( \\mu + p)} d ({\\bf k}).\n \\label{varphi_v-sol}\n\\end{eqnarray}\nThus, for $K = 0$ (but for general $\\Lambda$) we have\n\\begin{eqnarray}\n \\varphi_v ({\\bf k}, t) = C ({\\bf k}),\n \\label{varphi_v-sol2}\n\\end{eqnarray}\nwith the vanishing decaying solution. \nThe disappearance of the decaying \nsolution in Eq. (\\ref{varphi_v-sol2}) implies that the dominating decaying \nsolutions in Eqs. (\\ref{varphi_chi-sol},\\ref{v_chi-sol}) cancel out \nfor $K = 0$.\nIn fact, for $K = 0$, from Eqs. \n(\\ref{eq1},\\ref{eq2},\\ref{eq3},\\ref{eq6},\\ref{eq7}) we can derive\n\\begin{eqnarray}\n & & {c_s^2 H^2 \\over a^3 ( \\mu + p)} \\left[ {a^3 ( \\mu + p) \\over c_s^2 H^2}\n \\dot \\varphi_v \\right]^\\cdot\n + c_s^2 {k^2 \\over a^2} \\varphi_v = {\\rm stresses}.\n \\label{varphi_v-fluid-eq}\n\\end{eqnarray}\n{}For a pressureless case ($c_s^2 = 0$), instead of\nEq. (\\ref{varphi_v-fluid-eq}) we have $\\dot \\varphi_v = 0$;\nsee after Eq. (\\ref{CG-varphi-eq}).\nIn the large-scale limit, and ignoring the stresses, we have an integral\nform solution\n\\begin{eqnarray}\n & & \\varphi_v\n = C ({\\bf x}) - \\tilde D ({\\bf x})\n \\int^t_0 {c_s^2 H^2 \\over a^3 ( \\mu + p ) } dt.\n \\label{varphi_v-sol3}\n\\end{eqnarray}\nCompared with the solution in Eq. (\\ref{varphi_v-sol2}), the $\\tilde D$ \nterm in Eq. (\\ref{varphi_v-sol3}) is $c_s^2(k\/aH)^2$ order higher\nthan $d$ terms in the other gauge.\nTherefore, for $K = 0$, $\\varphi_v$ is conserved for the generally varying\nbackground equation of state, i.e., for general $p = p(\\mu)$.\nSince the solutions in \nEqs. (\\ref{varphi_chi-sol},\\ref{delta_v-sol},\\ref{v_chi-sol}) are valid \nin super-sound-horizon scale, the conservation property in \nEq. (\\ref{varphi_v-sol2}) is valid in all scales in the matter \ndominated era.\n\n{}For $\\varphi_\\delta$, by evaluating it in the comoving gauge we have\n\\begin{eqnarray}\n \\varphi_\\delta \\equiv \\varphi_v + {\\delta_v \\over 3 (1 + {\\rm w})}.\n\\end{eqnarray}\nUsing Eqs. (\\ref{varphi_v-sol},\\ref{delta_v-sol}) we have\n\\begin{eqnarray}\n \\varphi_\\delta ({\\bf k}, t)\n &=& C ({\\bf k}) \\left\\{ 1 + {k^2 \\over a^2}\n {1 \\over 12 \\pi G (\\mu + p)} \\left[ 1\n - {H \\over a} \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt\n \\right] \\right\\}\n + {k^2 \\over a^2} { H\/a \\over 12 \\pi G ( \\mu + p)} d ({\\bf k}).\n \\label{varphi_delta-sol}\n\\end{eqnarray}\nSince the higher order term ignored in Eq. (\\ref{varphi_v-sol}) \nis $c_s^2 (k\/aH)^2$ order higher, in the medium with $c_s \\sim 1 (\\equiv c)$,\nthe terms involving $(k\/a)^2\/(\\mu + p)$ are not necessarily valid.\nThus, to the leading order in the superhorizon scale we have\n\\begin{eqnarray}\n \\varphi_\\delta ({\\bf k}, t) = C ({\\bf k}).\n\\end{eqnarray}\nThus, in the superhorizon scale $\\varphi_\\delta$ is conserved \napparently considering $K$ and $\\Lambda$.\n[In the case of nonvanishing $K$ we may need to be \n careful in defining the superhorizon scale. \n Here, as the superhorizon condition we simply {\\it took} $k^2$ term \n to be negligible.\n In a hyperbolic (negatively curved) space with $K = -1$, \n since $R^{(3)} = {6 K \/ a^2}$, the distance \n $a\/\\sqrt{|K|}$ introduces a curvature scale: on smaller scales the \n space is near flat and on larger scales the curvature term dominates.\n In the unit of the curvature scale $k^2$ less than one ($\\sim |K|$)\n corresponds to the super-curvature scale, and $k^2$ larger than \n one corresponds to the sub-curvature scale. \n Thus, when we ignored the $k^2$ term compared with $|K|$ in \n a negatively curved space, we are considering the super-curvature scale.\n In other words, for the hyperbolic backgrounds $k^2$ takes continuous \n value with $k^2 \\ge 0$ where $k^2 \\ge 1$ corresponds to subcurvature \n mode whereas $0\\le k^2 <1 $ corresponds to supercurvature mode, and \n our large-scale limit took $k^2 \\rightarrow 0$.\n A useful study in the hyperbolic situation can be found \n in \\cite{Lyth-etal}.]\nHowever, in a medium with the negligible sound speed (like the matter dominated \nera), Eq. (\\ref{varphi_delta-sol}) is valid considering the $k^2$ terms.\nThus, as the scale enters the horizon in the matter dominated era \n$\\varphi_\\delta$ wildly changes its behavior and is {\\it no longer conserved \ninside the horizon};\ninside the horizon $\\varphi_\\delta$ is dominated by the $\\delta$ part\nwhich shows Newtonian behavior of $\\delta$ in most of the gauge conditions, \nsee Table 1.\n\n{}For $\\varphi_\\kappa$, by evaluating it in the uniform-density gauge\nand using Eq. (\\ref{eq2}), which gives \n$\\kappa_\\delta = (k^2 - 3K)\/(a^2 H) \\varphi_\\delta$, we have\n\\begin{eqnarray}\n \\varphi_\\kappa = \\varphi_\\delta\n \\left[ 1 + {k^2 - 3K \\over 12 \\pi G (\\mu + p) a^2} \\right]^{-1}.\n \\label{varphi_kappa-sol}\n\\end{eqnarray}\nThus, in the superhorizon scale and for $K = 0$, $\\varphi_\\kappa$ is conserved.\nAs in $\\varphi_\\delta$, in a medium with the negligible sound speed, \nthe solution in Eq. (\\ref{varphi_kappa-sol}) with Eq. (\\ref{varphi_delta-sol}) \nis valid considering the $k^2$ terms.\nThus, as the scale enters the horizon in the matter dominated era \n$\\varphi_\\kappa$\nalso changes its behavior and is {\\it no longer conserved inside the horizon}.\n\nNow we summarize:\nIn the superhorizon scale $\\varphi_v$ and $\\varphi_\\kappa$ are conserved for \n$K = 0$, while $\\varphi_\\delta$ is conserved considering the general $K$.\nIn the matter dominated era, for $K = 0$, $\\varphi_v$ is still conserved in the\nsuper-sound-horizon scale which virtually covers all scales,\nwhereas $\\varphi_\\delta$ and $\\varphi_\\kappa$ change their behaviors\nnear the horizon crossing epoch and are no longer conserved inside the horizon.\nIn this regard, we may say, $\\varphi_v$ is the {\\it best conserved quantity}.\nCuriously, although $\\varphi_\\chi$ is the variable most closely resembling\nthe Newtonian gravitational potential, it is not conserved for changing \nequation of state, see Eq. (\\ref{varphi_chi-sol});\nhowever, it is conserved independently of the horizon crossing in the\nmatter dominated era for $K = 0 = \\Lambda$, see Eq. (\\ref{Sol-w=const}).\n\nSimilar conservation properties of the curvature variable in certain gauge\nconditions remain true for models based on a minimally coupled scalar \nfield and even on classes of generalized gravity theories.\nIn the generalized gravity the uniform-field gauge is more suitable\nfor handling the conservation property, and the uniform-field gauge\ncoincides with the comoving gauge in the minimally coupled scalar field.\n{}Thorough studies of the minimally coupled scalar field\nand the generalized gravity, assuming $K = 0 = \\Lambda$, are made in \n\\cite{MSF-GGT,MSF-GGT-2} and summarized in \\cite{GRG-MSF-GGT}.\n\n\n\\subsection{Multi-component situation}\n \\label{sec:Multi}\n\nWe consider the energy-momentum tensor composed of multiple components as\n\\begin{eqnarray}\n T_{ab} = \\sum_{(l)} T_{(l)ab}, \\quad\n T^{\\;\\;\\;\\; b}_{(i)a;b} \\equiv Q_{(i)a}, \\quad\n \\sum_{(l)} Q_{(l)a} = 0,\n \\label{Tab-multi}\n\\end{eqnarray}\nwhere $i,l (= 1,2, \\dots, n)$ indicates $n$ fluid components, and\n$Q_{(i)a}$ considers possible mutual interactions among fluids.\nSince we are considering the scalar-type perturbation we decompose \nthe $Q_{(i)a}$ in the following way:\n\\begin{eqnarray}\n Q_{(i)0} \\equiv - a (1 + \\alpha) Q_{(i)}, \\quad\n Q_{(i)} \\equiv \\bar Q_{(i)} + \\delta Q_{(i)}, \\quad\n Q_{(i)\\alpha} \\equiv J_{(i),\\alpha}.\n\\end{eqnarray}\nThe scalar-type fluid quantities in Eq. (\\ref{Tab}) can be considered \nas the collective ones which are related to the individual component as:\n\\begin{eqnarray}\n & & \\bar \\mu = \\sum_{(l)} \\bar \\mu_{(l)}, \\quad\n \\bar p = \\sum_{(l)} \\bar p_{(l)}; \\quad\n \\delta \\mu = \\sum_{(l)} \\delta \\mu_{(l)}, \\quad\n \\delta p = \\sum_{(l)} \\delta p_{(l)}, \n \\nonumber \\\\\n & & (\\mu + p) v = \\sum_{(l)} (\\mu_{(l)} + p_{(l)} ) v_{(l)}, \\quad\n \\sigma = \\sum_{(l)} \\sigma_{(l)}.\n \\label{fluid-multi}\n\\end{eqnarray}\n\n{}For the background Eq. (\\ref{BG-eqs}) remains valid for the collective fluid\nquantities.\nAn additional equation follows from the individual\nenergy-momentum conservation in Eq. (\\ref{Tab-multi}) as\n\\begin{eqnarray}\n \\dot \\mu_{(i)} + 3 H \\left( \\mu_{(i)} + p_{(i)} \\right) = Q_{(i)}.\n \\label{BG-multi}\n\\end{eqnarray}\n{}For the perturbations Eqs. (\\ref{eq1}-\\ref{eq7}) remain valid for the \ncollective fluid quantities. \nThe additional equations we need follow from the individual\nenergy-momentum conservation in Eq. (\\ref{Tab-multi}).\n{}From $T^{\\;\\;\\;\\; b}_{(i)0;b} = Q_{(i)0}$ and using Eq. (\\ref{eq1})\nwe have the energy conservation of the fluid components \n\\begin{eqnarray}\n \\delta \\dot \\mu_{(i)} + 3 H \\left( \\delta \\mu_{(i)}\n + \\delta p_{(i)} \\right)\n = - {k \\over a} \\left( \\mu_{(i)} + p_{(i)} \\right) v_{(i)}\n + \\dot \\mu_{(i)} \\alpha\n + \\left( \\mu_{(i)} + p_{(i)} \\right) \\kappa + \\delta Q_{(i)}.\n \\label{eq8}\n\\end{eqnarray}\n{}From $T^{\\;\\;\\;\\; b}_{(i)\\alpha;b} = Q_{(i)\\alpha}$ we have\nthe momentum conservation of the fluid components\n\\begin{eqnarray}\n \\dot v_{(i)} + \\left[ H \\left( 1 - 3 c_{(i)}^2 \\right)\n + { 1 + c_{(i)}^2 \\over \\mu_{(i)} + p_{(i)} } Q_{(i)} \\right] v_{(i)}\n = {k \\over a} \\left[ \\alpha\n + {1 \\over \\mu_{(i)} + p_{(i)}} \\left( \\delta p_{(i)}\n - {2 \\over 3} {k^2 - 3 K \\over a^2} \\sigma_{(i)}\n - J_{(i)} \\right) \\right].\n \\label{eq9}\n\\end{eqnarray}\nBy adding Eqs. (\\ref{eq8},\\ref{eq9}) over the components we have\nEqs. (\\ref{eq6},\\ref{eq7}).\nIn the multi-component situation \nEqs. (\\ref{eq1}-\\ref{eq7},\\ref{eq8},\\ref{eq9}) are the complete set \nexpressed in a gauge-ready form.\nIf we decompose the pressure as\n$\\delta p_{(i)} = c_{(i)}^2 \\delta \\mu_{(i)} + e_{(i)}$ with\n$c_{(i)}^2 \\equiv \\dot p_{(i)}\/\\dot \\mu_{(i)}$, comparing with\nEq. (\\ref{p-decomposition}) we have\n\\begin{eqnarray}\n e = \\delta p - c_s^2 \\delta \\mu\n = \\sum_{(l)} \\left[ \\left( c_{(l)}^2 - c_s^2 \\right) \\delta \\mu_{(l)}\n + e_{(l)} \\right].\n \\label{e}\n\\end{eqnarray}\n\nUnder the gauge transformation, from Eq. (20) in \\cite{PRW}, we have:\n\\begin{eqnarray}\n & & \\delta \\tilde \\mu_{(i)} = \\delta \\mu_{(i)} - \\dot \\mu_{(i)} \\xi^t, \\quad\n \\delta \\tilde p_{(i)} = \\delta p_{(i)} - \\dot p_{(i)} \\xi^t, \\quad\n \\tilde v_{(i)} = v_{(i)} - {k \\over a} \\xi^t, \n \\nonumber \\\\\n & & \\tilde J_{(i)} = J_{(i)} + Q_{(i)} \\xi^t, \\quad\n \\delta \\tilde Q_{(i)} = \\delta Q_{(i)} - \\dot Q_{(i)} \\xi^t.\n \\label{GT-2}\n\\end{eqnarray}\nThus, we have the following additional gauge conditions:\n\\begin{eqnarray}\n \\delta \\mu_{(i)} \\equiv 0, \\quad\n \\delta p_{(i)} \\equiv 0, \\quad\n v_{(i)}\/k \\equiv 0, \\quad\n {\\rm etc}.\n \\label{Gauges-2}\n\\end{eqnarray}\nAny one of these gauge conditions also fixes the temporal gauge condition\ncompletely.\nStudies of the multi-component situations\ncan be found in \\cite{KS,PRW,KS-2}.\n\n\n\\subsection{Rotation and Gravitational Wave}\n \\label{sec:rot-GW}\n\n{}For the vector-type perturbation, using Eqs. (\\ref{metric},\\ref{Tab}),\nwe have:\n\\begin{eqnarray}\n & & 8 \\pi G T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha}\n = - {\\Delta + 2 K \\over 2 a^2} \\left( B_\\alpha + a \\dot C_\\alpha \\right),\n \\label{rot-1} \\\\\n & & 8 \\pi G \\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n = {1 \\over 2 a^3} \\left\\{ a^2 \\left[ B^\\alpha_{\\;\\;|\\beta}\n + B_\\beta^{\\;\\;|\\alpha} + a \\left( C^\\alpha_{\\;\\;|\\beta}\n + C_\\beta^{\\;\\;|\\alpha} \\right)^\\cdot \\right] \\right\\}^\\cdot,\n \\label{rot-2} \\\\\n & & {1 \\over a^3} \\left( a^4 T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha} \\right)^\\cdot\n = - \\delta T^{(v)\\beta}_{\\;\\;\\;\\;\\;\\alpha|\\beta},\n \\label{rot-3}\n\\end{eqnarray}\nwhere Eq. (\\ref{rot-3}) follows from $T^b_{\\alpha ; b} = 0$, and \napparently Eqs. (\\ref{rot-1},\\ref{rot-2}) follow from the Einstein equations.\nWe can show that Eq. (\\ref{rot-3}) follows from Eqs. (\\ref{rot-1},\\ref{rot-2}).\nWe introduce a gauge-invariant velocity related variable \n$V^{(\\omega)}_\\alpha ({\\bf x}, t)$ as \n$T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha} \\equiv (\\mu + p) V^{(\\omega)}_\\alpha$.\nWe can show $V^{(\\omega)}_\\alpha \\propto a \\omega$ where $\\omega$ \nis the amplitude of vorticity vector, see Sec. 5 of \\cite{PRW}.\nThus, ignoring the anisotropic stress in the RHS of Eq. (\\ref{rot-3})\nwe have the conservation of angular momentum as\n\\begin{eqnarray}\n {\\rm Angular \\; Momentum} \\sim a^3 \\left( \\mu + p \\right)\n \\times a \\times V^{(\\omega)}_\\alpha ({\\bf x}, t)\n = {\\rm constant \\; in \\; time},\n\\end{eqnarray}\nwhich is valid for general $K$, $\\Lambda$, and $p(\\mu)$ in general scale.\n\n{}For the tensor-type perturbation, using Eqs. (\\ref{metric},\\ref{Tab})\nin the Einstein equations, we have\n\\begin{eqnarray}\n 8 \\pi G \\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n = \\ddot C^\\alpha_\\beta + 3 H \\dot C^\\alpha_\\beta\n - {\\Delta - 2 K \\over a^2} C^\\alpha_\\beta.\n \\label{GW-eq}\n\\end{eqnarray}\nIn the large-scale limit, assuming $K = 0$ and ignoring the anisotropic\npressure Eq. (\\ref{GW-eq}) has a general integral form solution\n\\begin{eqnarray}\n C^\\alpha_\\beta ({\\bf x}, t)\n = C^{\\;\\alpha}_{1\\beta} ({\\bf x}) - D^\\alpha_\\beta ({\\bf x})\n \\int^t {1 \\over a^3} dt,\n\\end{eqnarray}\nwhere $C^{\\;\\alpha}_{1\\beta}$ and $D^\\alpha_\\beta$ are integration constants\nfor relatively growing and decaying solutions, respectively.\nThus, ignoring the transient term the amplitude of the gravitational\nwave is conserved in the super-horizon scale considering general\nevolution of the background equation of state.\n\nThe conservation of angular momentum and the equation for the\ngravitational wave were first derived in \\cite{Lifshitz}.\nEvolutions in the multi-component situation were considered in\nSec. 5 of \\cite{PRW}.\n\n\n\\section{Discussions}\n \\label{sec:Discussion}\n\nWe would like to make comments on related works in several {\\it textbooks}:\nEq. (15.10.57) in \\cite{Weinberg}, Eq. (10.118) in \\cite{Peebles-2},\nEq. (11.5.2) in \\cite{Coles}, Eq. (8.52) in \\cite{Moss},\nand problem 6.10 in \\cite{Padmanabhan} are in error.\nAll these errors are essentially about the same point involved with\na fallible algebraic mistake in the synchronous gauge.\nThe correction in the case of \\cite{Weinberg} was made in \\cite{GRG}:\nin a medium with a nonvanishing pressure the equation for the density \nfluctuation in the synchronous gauge becomes third order because of \nthe presence of a gauge mode in addition to the physical growing \nand decaying solutions which is true even in the large-scale limit.\nThe truncated second order equation in \\cite{Weinberg,Moss} \npicks up a gauge mode instead of the physical decaying solution in the \nsynchronous gauge.\nThe errors in \\cite{Peebles-2,Padmanabhan} are based on imposing \nthe synchronous gauge and the comoving gauge simultaneously, and thus \nhappen to end up with the same truncated second order equation \nas in \\cite{Weinberg}.\nIn a medium with nonvanishing pressure one cannot impose the two gauge \nconditions simultaneously (even in the large-scale limit). \nIn Sec. 11.5 of \\cite{Coles} the authors proposed a simple modification\nof the Newtonian theory which again happens to end up with the same incorrect \nequation as in the other books.\nIn the Appendix we elaborate our point.\nFor comments on other related errors in the literature, see Sec. 3.10 in \n\\cite{Ideal} and \\cite{GRG}.\n\nWe would like to remark that these errors found in the synchronous\ngauge are not due to any esoteric aspect of the gauge choice in the \nrelativistic perturbation analyses.\nAlthough not fundamental, the number of errors found in the literature\nconcerning the synchronous gauge seem to indicate the importance of using \nthe proper gauge condition in handling the problems. \n{}For our own choice of the preferred gauge-invariant variables \nsuitable for handling the hydrodynamic perturbation and consequent analyses, \nsee Sec. \\ref{sec:Hydrodynamics}; the reasons for such choices are made in \nSec. \\ref{sec:Six-gauges}.\nWe wish to recall, however, that although one may need to do more algebra \nin tracing the remnant gauge mode, even the synchronous gauge condition\nis adequate for handling many problems as was carefully done in the \noriginal study by Lifshitz in 1946 \\cite{Lifshitz}.\n\nIn this paper we have tried to identify the variables in the relativistic\nperturbation analysis which reproduce the correct Newtonian behavior\nin the pressureless limit.\nIn the first part we have shown that\n$\\delta$ in the comoving gauge ($\\delta_v$) and $v$ and $\\varphi$ \nin the zero-shear gauge ($v_\\chi$ and $\\varphi_\\chi$) \nshow the same behavior as the corresponding Newtonian \nvariables {\\it in general scales}.\nIn fact, these results have already been presented in \\cite{MDE}. \nAlso, various general and asymptotic solutions for every variable\nin the pressureless medium are presented in the Tables of \\cite{MDE}.\nCompared with \\cite{MDE}, in Sec. \\ref{sec:Six-gauges} we tried to reinforce \nthe correspondence by explicitly showing the second order differential \nequations in several gauge conditions.\nWe have also added some additional insights gained after publishing \\cite{MDE}.\nThe second part contains some original results.\nUsing the gauge-invariant variables $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ \nwe write the relativistic hydrodynamic cosmological perturbation equations in \nEqs. (\\ref{P-1}-\\ref{P-4}); actually, these equations \nare also known in \\cite{Bardeen-1980}.\nThe new results in this paper are the compact way of deriving the\ngeneral large-scale solutions in Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}), \nand the clarification of the general large-scale conservation property of\n$\\varphi_v$ in Eq. (\\ref{varphi_v-sol}).\n\nThe underlying mathematical or physical reasons for \nthe variables $\\delta_v$, $\\varphi_\\chi$, $v_\\chi$, and $\\varphi_v$ \nhaving the distinguished behaviors, compared with many other available\ngauge-invariant combinations, may still deserve further investigation.\n\n\n\\subsection*{Acknowledgments}\n\nWe thank Profs. P. J. E. Peebles and A. M\\'esz\\'aros for useful discussions\nand comments, and Prof. R. Brandenberger for interesting correspondences.\nJH was supported by the KOSEF, Grant No. 95-0702-04-01-3 and\nthrough the SRC program of SNU-CTP.\nHN was supported by the DFG (Germany) and the KOSEF (Korea).\n\n\n\\section*{Appendix: Common errors in the synchronous gauge}\n\nIn the following we correct a minor confusion in the literature\nconcerning perturbation analyses in the synchronous gauge.\nThe argument is based on \\cite{GRG}.\n{}For simplicity, we consider a situation with $K = 0 = \\Lambda$,\n$e = 0 = \\sigma$ and ${\\rm w} = {\\rm constant}$ (thus $c_s^2 = {\\rm w}$).\n\nThe equation for the density perturbation in the {\\it comoving gauge}\nis given in Eq. (\\ref{delta-eq}).\nIn our case we have\n$$\n \\ddot \\delta_v + ( 2- 3 {\\rm w}) H \\dot \\delta_v\n + \\left[ c_s^2 {k^2 \\over a^2}\n - 4 \\pi G \\mu (1 - {\\rm w}) ( 1 + 3 {\\rm w} ) \\right] \\delta_v = 0.\n \\eqno{(A1)}\n$$\nThe solution in the super-sound-horizon scale is presented in\nEq. (\\ref{Sol-w=const}).\n\nIn the {\\it synchronous gauge}, from Eqs. (\\ref{eq5}-\\ref{eq7}) we have\n(the subindex $\\alpha$ indicates the synchronous gauge variable)\n$$\n \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha\n + \\left[ c_s^2 {k^2 \\over a^2}\n - 4 \\pi G \\mu ( 1+ {\\rm w} ) ( 1 + 3 {\\rm w} ) \\right] \\delta_\\alpha\n + 3 {\\rm w} ( 1 + {\\rm w}) {k \\over a} H v_\\alpha = 0,\n \\eqno{(A2)}\n$$\n$$\n \\dot v_\\alpha + ( 1- 3 {\\rm w}) H v_\\alpha\n - {k \\over a} {{\\rm w} \\over 1 + {\\rm w}} \\delta_\\alpha = 0.\n \\eqno{(A3)}\n$$\nNotice that for ${\\rm w} \\neq 0$ these two equations are generally coupled\neven in the large-scale limit.\n{}From these two equations we can derive a third order differential equation\nfor $\\delta_\\alpha$ as\n$$\n \\delta^{\\cdot\\cdot\\cdot}_\\alpha\n + {11 - 3 {\\rm w} \\over 2} H \\ddot \\delta_\\alpha\n + \\left( {5 - 24 {\\rm w} - 9 {\\rm w}^2 \\over 2} H^2\n + c_s^2 {k^2 \\over a^2} \\right) \\dot \\delta_\\alpha\n$$\n$$\n + \\left[ {3 \\over 4} \\left( 1 + {\\rm w} \\right)\n \\left( 1 + 3 {\\rm w} \\right) \\left( -1 + 9 {\\rm w} \\right) H^3\n + {3 \\over 2} \\left( 1 + {\\rm w} \\right) H c_s^2 {k^2 \\over a^2}\n \\right] \\delta_\\alpha = 0.\n \\eqno{(A4)}\n$$\nIn the large-scale limit the solutions are\n$$\n \\delta_\\alpha \\quad \\propto \\quad\n t^{{ 2(1+ 3{\\rm w}) \\over 3(1 + {\\rm w})}}, \\quad\n t^{{9{\\rm w} - 1 \\over 3( 1+ {\\rm w})}}, \\quad\n t^{-1}.\n \\eqno{(A5)}\n$$\nIn Appendix B of \\cite{GRG} we showed that\nthe third solution with $\\delta_\\alpha \\propto t^{-1}$\nis nothing but a gauge mode for a medium with ${\\rm w} \\neq 0$.\n[As mentioned before, the combination $\\delta_\\alpha$ is not gauge-invariant.\n {}From Eq. (\\ref{GT}) we have\n $\\delta_\\alpha \\equiv \\delta + 3 H ( 1 + {\\rm w}) \\int^t \\alpha dt$\n and the lower bound of the integration gives rise to the gauge mode \n which is proportional to $t^{-1}$.]\n{}For a pressureless case the physical decaying solution also behaves \nas $t^{-1}$ and the second solution in Eq. (A5) is invalid,\nsee Sec. 4 in \\cite{GRG}.\nIn Eq. (B5) of \\cite{GRG} we derived the relation between solutions\nin the two gauges explicitly; the growing solutions are the same in both gauges\nwhereas the decaying solutions differ by a factor\n$(k \/ aH)^2 \\propto t^{2(1 + 3 {\\rm w}) \\over 3 (1 + {\\rm w})}$.\n\nNow, we would like to point out that, in a medium with ${\\rm w} \\neq 0$,\none cannot ignore the last term in Eq. (A2)\neven in the large-scale limit.\nIf we {\\it incorrectly} ignore the last term in Eq. (A2) we recover\nthe {\\it wrong equation} in the textbooks mentioned in \nSec. \\ref{sec:Discussion} which is\n$$\n \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha\n + \\left[ c_s^2 {k^2 \\over a^2}\n - 4 \\pi G \\mu ( 1 + {\\rm w} ) ( 1 + 3 {\\rm w} ) \\right] \\delta_\\alpha\n = 0.\n \\eqno{(A6)}\n$$\nIn the large-scale we have solutions\n$$\n \\delta_\\alpha \\quad \\propto \\quad\n t^{{ 2(1+ 3{\\rm w}) \\over 3(1 + {\\rm w})}}, \\quad\n t^{-1}.\n \\eqno{(A7)}\n$$\nBy ignoring the last term in Eq. (A2), in the large-scale\nlimit we happen to recover the fictitious gauge mode under the price of losing\nthe physical decaying solution [the second solution in Eq. (A5)].\nThus, in the large-scale limit one cannot ignore the last term in Eq. (A2)\nin a medium with a general pressure; the reason is obvious if we see\nEq. (A4).\nAlso, one cannot impose both the synchronous gauge condition and the\ncomoving gauge condition simultaneously.\nIf we simultaneously impose such two conditions,\nthus setting $v \\equiv 0 \\equiv \\alpha$, from Eq. (\\ref{eq7}) we have\n$$\n 0 = {k \\over a \\left( 1 + {\\rm w} \\right) } \\left( c_s^2 \\delta\n + {e \\over \\mu} - {2\\over 3} {k^2 - 3K \\over a^2} {\\sigma \\over \\mu}\n \\right).\n \\eqno{(A8)}\n$$\nThus, for $e = 0 = \\sigma$ and medium with the nonvanishing pressure\nwe have $\\delta = 0$ which is a meaningless system;\nthis argument remains valid even in the large-scale limit.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{ Introduction }\nIt has long been a problem that the $\\Delta I=3\/2$ \n$K^+\\to\\pi^+\\pi^0$ amplitude calculated in quenched lattice QCD is \nabout a factor two too large compared to the experimental value\\cite{BSReview}.\nIn this article we report a new study of this problem,\nincorporating various theoretical and technical advances \nin recent years for analysis.\nIn particular we discuss in detail how one-loop corrections of \nchiral perturbation theory (CHPT) recently calculated\\cite{ONECPTH} affect\nphysical predictions for the decay amplitude from lattice QCD simulations. \n\nOur simulation is carried out in quenched lattice QCD employing \nthe standard plaquette action for gluons at $\\beta=6.1$ and the Wilson action for quarks. \nTwo lattice sizes, $24^3\\times 64$ and $32^3\\times 64$, are employed.\nWe take up, down and strange quarks to be degenerate,\nand make measurements at four values of the common hopping parameter,\n$\\kappa = 0.1520$, $0.1530$, $0.1540$ and $0.1543$, which correspond to\n$m_\\pi\/m_\\rho = 0.797$, $0.734$, $0.586$ and $0.515$.\n\\section{ Extraction of decay amplitude }\nThe 4-quark operator most relevant for the \n$\\Delta I=3\/2$ $K\\to\\pi\\pi$ decay is\n$Q_+ = [ ( \\bar{s} d )_L\n ( \\bar{u} u )_L\n + ( \\bar{s} u )_L\n ( \\bar{u} d )_L ] \/ 2$.\nWe extract the decay amplitude from the 4-point correlation function defined by\n$M_Q = \\langle 0 | W_+ W_0 Q_{+}(t) W_K | 0 \\rangle$ \nwhere $W_{0,+,K}$ are wall sources for $\\pi^0$, $\\pi^+$ and $K^+$. \nIn our calculations on a lattice of a temporal size $T=64$, \nthe walls are placed at the time slices $t_{K^+}=4$, $t_{\\pi^+}=59$ and $t_{\\pi^0}=60$.\nThe mesons are all created at rest, and the 4-quark operator $Q_+$ is \nprojected to zero spatial momentum.\n\nWe can use two types of factors to remove the normalization factors in $M_Q$.\nIf we define\n\\begin{eqnarray}\n\\lefteqn{ M_W =\n \\langle 0 | W_0 \\pi^{0} ( t ) | 0 \\rangle\n \\langle 0 | W_+ \\pi^{+} ( t ) | 0 \\rangle\n \\langle 0 | K(t) W_K | 0 \\rangle ,} && \\nonumber \\\\\n\\lefteqn{ M_P = \n \\langle 0 | K(t) W_K | 0 \\rangle \n \\langle 0 | W_+ W_0 \\pi^{+} ( t ) \\pi^{0} ( t ) | 0 \\rangle , } && \n\\end{eqnarray}\nwe find\n$R_W \\equiv M_Q \/ M_W$\n$\\sim \\langle \\pi^+\\pi^0 | Q_+ | K^+ \\rangle$\n$ \/ \\langle \\pi | \\pi | 0 \\rangle^3 \\cdot {\\rm exp}(t-t_+)\\Delta$\nand\n$R_P \\equiv M_Q \/ M_P$ \n$\\sim \\langle \\pi^+\\pi^0 | Q_+ | K^+ \\rangle \/ \\langle \\pi | \\pi | 0 \\rangle^3$\nfor $t_K\\ll t\\ll t_+, t_0$, where $\\Delta=m_{\\pi\\pi}-2m_\\pi$ is the \nmass shift of the 2-pion state due to finite lattice size \neffects~\\cite{LUSHER}.\n \nIn Fig.~\\ref{RWFRPF} we plot $\\langle \\pi | \\pi | 0 \\rangle^3 \\cdot R_W$ and \n$\\langle \\pi | \\pi | 0\\rangle^3 \\cdot R_P$ at \n$\\kappa=0.1530$ as a function of time $t$ of the weak operator.\nWe clearly observe a non-vanishing slope for $R_W$, while $R_P$ exhibits a \nplateau as expected.\nThe decay amplitude can be obtained by fitting $R_W$ to a single exponential \nor $R_P$ to a constant. We find the results to be mutually consistent \nwithin the statistical error.\n\\begin{figure}[t]\n\\vspace*{-0.2cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{L24L32all.1.eps}}\n\\vspace*{-1cm}\n\\caption{\\label{RWFRPF}\n$R_W \\cdot \\langle \\pi | \\pi | 0 \\rangle^3$ and $R_P\\langle \\pi | \\pi | 0 \\rangle^3$\nat $\\kappa=0.153$.\nOpen and filled circles refer to data for $24^3$ and $32^3$ lattices.\n}\n\\vspace*{-0.7cm}\n\\end{figure}\n\\begin{figure}[t]\n\\vspace{-0.6cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{o00210o11210.00.eps}}\n\\vspace{-1.0cm}\n\\caption{ \\label{COMPARISON}\nComparison of our results normalized by the experimental value obtained \nwith tree-level CHPT relation for $q^*=\\pi\/a$ at $\\beta=6.1$ \nwith those of previous work{\\protect\\cite{BSKpipi}} at $\\beta=6.0$ (crosses).\nOpen symbols refer to results with the traditional normalization \n$\\protect\\sqrt{2\\kappa}$ and no tadpole improvement, \nand filled ones are those with the KLM normalization and tadpole improvement.\n}\n\\vspace{-0.7cm}\n\\end{figure}\n\nThe one-loop renormalization factor relating the operator $Q_+$ on the lattice \nand that in the continuum was obtained in Refs.~\\cite{Z-M,Z-BS}. \nWe set $q^*=1\/a$ or $\\pi\/a$ as the matching point and employ tadpole-improved\nperturbation theory with \nthe mean-field improved $\\overline{\\rm MS}$ coupling constant\\cite{TPI}. \nQuark fields are normalized with the KLM factor~\\cite{KLM}, \n$\\sqrt{1-3\\kappa\/4\\kappa_c}$.\n\\section{ Result }\nIn earlier calculations tree-level chiral perturbation theory (CHPT) \nformula was used to obtain the physical amplitude $A^{phys}$ from the \nlattice counterpart $A^{lat}$. The formula takes the form \n$ A^{phys} = ( m_K^2 - m_\\pi^2 )\/( 2 M_\\pi^2 ) \\cdot A^{lat}$\nwhere $m_K=497 {\\rm MeV}$ and $m_\\pi=136 {\\rm MeV}$ are physical masses, \nand $M_\\pi$ is the degenerate mass of $K$ and $\\pi$ mesons on the lattice.\nClearly the physical amplitude should be independent of the \nlattice mass $M_\\pi$ if the matching formula is exact.\n\nIn Fig.~\\ref{COMPARISON} we compare results for the physical decay amplitude \nfrom a previous work\\cite{BSKpipi} carried out at $\\beta=6.0$ on a $24^3$ \nspatial lattice with those of our simulation at $\\beta=6.1$. \nOur values obtained with the conventional quark normalization $\\sqrt{2\\kappa}$\n(crosses and open squares) are consistent with their results,\nbut are larger than the experimental result roughly by a factor two.\n\nLet us also note with our results (circles and squares) \nthat (i) the KLM normalization and tadpole improvement of \nthe operator have a significant effect on the amplitude, (ii) there is \na clear dependence of the amplitude on the lattice meson mass $M_\\pi$, \nand (iii) a significant finite-size effect is observed between the two \nlattice sizes.\nThese features show that the tree-level CHPT is inadequate to extract \nthe physical amplitude from lattice results.\n\nIn Fig.~\\ref{FRHO} we show how predictions for the physical amplitude \nchange if we apply the one-loop formula of CHPT~\\cite{ONECPTH} to our results.\nHere $\\Lambda^q$ and $\\Lambda^{cont}$ are the cutoffs of CHPT for quenched and full theory.\nIn converting to physical values we ignore effects of $O(p^4)$ contact terms \nof the CHPT Lagrangian since their values are not well known.\nAn interesting point is that a size dependence seen with \nthe tree-level analysis in Fig.~\\ref{COMPARISON} is absent.\nAnother important point is that the magnitude of the amplitude decreases \nby $30-40$\\% over the range of meson mass covered in our simulation.\nThe amplitude depends significantly on the quenched lattice cutoff \n$\\Lambda^q$, in particular toward larger values of $M_\\pi$.\n\n\\begin{figure}\n\\vspace*{-0.6cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{o11210.1113.eps}}\n\\vspace*{-1cm}\n\\caption{ \\label{FRHO} \nDecay amplitude normalized by experimental value for $q^*=\\pi\/a$ \nobtained with one-loop CHPT for $\\Lambda^{cont} = 770{\\rm MeV}$.\nCircles and squares refer to data for $24^3$ and $32^3$ spatial sizes. \nOpen symbols are for $\\Lambda^{q}=770{\\rm MeV}$ and filled ones for \n$1\\ {\\rm GeV}$.\n}\n\\vspace*{-7mm}\n\\end{figure}\nOur results in Fig.~\\ref{FRHO} show that a sizable meson mass dependence still remains.\nThis may be attributed to $O(p^4)$ contact terms which were not taken into account.\nIf we assume that the contact terms of the full theory are as small \nas suggested by the phenomenological analysis~\\cite{BK},\nwe can estimate the physical amplitude by a linear extrapolation of $M_\\pi^2$ to the chiral limit.\nIn Table~\\ref{FINALFIT} we list results for several choices of the cutoff\nand the operator matching point $q^*$.\nAs is expected from Fig.~\\ref{FRHO}, variation of results on the \nchoice of the cutoff parameters is small, being of the order of 10\\%.\nAllowing for this uncertainty, the values in Table~\\ref{FINALFIT}\nare consistent with the experimental value of $10.4\\times 10^{-3} {\\rm GeV}^3$.\n\\section{ Conclusions }\nThe present study has shown that the quenched result for the \n$K^+\\to\\pi^+ \\pi^0 $ decay amplitude agrees with experiment much better \nthan previously thought, especially if the chiral limit is taken \nfor the lattice meson mass.\nAway from the chiral limit, the amplitude depends significantly on the cutoff scales of CHPT, however. \nThis problem is largely ascribed to a mismatch of chiral logarithms \nbetween the quenched and full theories\\cite{ONECPTH}. \nIn order to obtain $K^+\\to\\pi^+ \\pi^0 $ decay amplitude free from this uncertainty,\nwe need simulations in full QCD.\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{0.2pc}\n\\begin{tabular}{llllll}\n\\hline\n\\hline\n&&\\multicolumn{4}{c}{$C_+\\langle \\pi\\pi |Q_+| K \\rangle (\\times 10^{-3} {\\rm GeV}^3)$}\\\\\n$\\Lambda^{cont}$ & $\\Lambda^{q}$ & \\multicolumn{2}{c}{$24^3$}&\\multicolumn{2}{c}{$32^3 $}\\\\\n({\\small GeV}) & ({\\small GeV}) &$q^*=\\frac{1}{a}$&$q^*=\\frac{\\pi}{a}$&$q^*=\\frac{1}{a}$&$q^*=\\frac{\\pi}{a}$\\\\[1mm]\n\\hline\n0.77 & 0.77 & $ 9.3(19)$&$10.2(21)$ & $8.9(17)$&$ 9.7(19)$ \\\\\n0.77 & 1.0 & $ 9.4(13)$&$10.3(14)$ & $8.8(11)$&$9.6(12)$ \\\\\n1.0 & 0.77 & $10.3(21)$&$11.3(23)$ & $9.8(19)$&$10.7(21)$ \\\\\n1.0 & 1.0 & $10.4(14)$&$11.4(15)$ & $9.7(12)$&$10.6(13)$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\vspace*{3mm}\n\\caption{\\label{FINALFIT}\nResults of linear extrapolation of the decay amplitude to $M_\\pi^2=0$. \nStatistical and extrapolation errors are combined.\nThe experimental value is $10.4\\times 10^{-3} {\\rm GeV}^3$.\n}\n\\vspace*{-7mm}\n\\end{table}\n\\hfill\\break\nThis work is supported by the Supercomputer \nProject (No.~97-15) of High Energy Accelerator Research Organization (KEK),\nand also in part by the Grants-in-Aid of \nthe Ministry of Education (Nos. 08640349, 08640350, 08640404,\n09246206, 09304029, 09740226).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}