diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmxhx" "b/data_all_eng_slimpj/shuffled/split2/finalzzmxhx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmxhx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nUnderstanding the microscopic origin of black hole thermodynamics is one of the most important outstanding problem in theoretical physics.\nRecently Ref.\\cite{Morita:2013wfa} proposed the following microscopic description of the near extremal black branes in the Einstein-Maxwell-Dilaton system. \nThis system admits the extreme brane solution in which no force works between the parallel branes if we tune the dilaton coupling suitably.\nIn case these branes are nearly parallel or slowly moving, they start interacting with each other and\nthe low energy effective action for these branes is given as\n\\begin{align}\nS=\\int dt \\left(\n\\sum_{i=1}^{N} \\frac{m}{2} \\vect{v}{2}_i+\n\\sum_{i\\neq j}^{N} \\frac{\\kappa^2_{4} m^2}{16\\pi} \\frac{\\vect{v}{4}_{ij}}{\\vec{r}_{ij}}\n+\\cdots\\right).\n\\label{single-brane}\n\\end{align}\nHere we consider 0-branes in four dimensions as a simple example. $N$ is the number of the branes, $\\vec{v}_i$ is velocity of the $i$-th brane, $m$ is mass of the brane and $\\kappa_{4}$ is the gravitational coupling.\nAlso $\\vec{r}_{ij}\\equiv \\vec{r}_i-\\vec{r}_j$ and $\\vec{v}_{ij}\\equiv \\vec{v}_i-\\vec{v}_j$ denote the relative position and the relative velocity of the $i$-th and $j$-th branes.\nThe first term is the ordinary non-relativistic kinetic term and the second term is the interactions between the branes which vanish when the branes are relatively static.\n\nIf we assume that the branes compose a bound state due to the interactions, we can estimate the thermodynamical properties of the bound state by applying the virial theorem to the effective action (\\ref{single-brane}).\nIn \\cite{Morita:2013wfa}, we found that the branes are strongly coupled in the bound state and the thermodynamical properties are coincident with the corresponding black brane thermodynamics in the near extremal regime (up to unfixed numerical factors).\nIt suggests that the black hole thermodynamics may be explained via the effective action (\\ref{single-brane}) microscopically.\nWe call this conjecture ``$p$-soup proposal\".\n\n\nThis proposal works in the various BPS branes in superstring and M-theory, and we can explain the black hole thermodynamics of the black brane solutions through the effective actions for these branes similar to (\\ref{single-brane}) \\cite{Wiseman:2013cda, Morita:2013wla, Morita:2014ypa}.\n(The related studies have been done in \\cite{Horowitz:1997fr,Li:1997iz,Banks:1997tn,Li:1998ci,Smilga:2008bt}.)\nMoreover, this proposal works even in the D1-D5(-P) system \\cite{Morita:2014cfa}.\nSince this system \\cite{Callan:1996dv,David:2002wn} involves the different species of the branes and the supersymmetry is less than the single brane case, it is non-trivial that the $p$-soup proposal works in such a system.\n\nThis motivates us to investigate general multiple species of branes in the Einstein-Maxwell-Dilaton system.\nWe set these multiple species of branes so that the forces between the branes vanish when they are static. \nIt happens when the branes satisfy so-called ``intersecting rule'' \\cite{Argurio:1997gt}, and the D1-D5(-P) system also satisfies it.\nThen we derive the effective action for these branes similar to (\\ref{single-brane}), and, by analyzing this action, we find that the bound state of these branes exhibits thermodynamical properties corresponding to the black hole thermodynamics.\nThus the $p$-soup proposal works in this system, and we believe that it captures the fundamental natures of the black hole microstate.\n\n\n\nThis paper is organized as follows.\nIn Sec.\\,\\ref{sec:setup} we review Einstein-Maxwell-Dilaton theory and clarify our setup of intersecting brane systems.\nIn Sec.\\,\\ref{sec:twobrane} we discuss intersecting brane systems of only two species of branes. This simple example would be helpful to understand our discussion without complicating the story.\nThe general intersecting brane systems are discussed in Sec.\\,\\ref{sec:general}, but the details of calculation are shown in \\ref{app:eff}.\nSec.\\,\\ref{sec-conclusion} is the discussions including possible applications to string theory.\nThe corresponding black branes in supergravity and their thermodynamics are discussed in \\ref{app:thermo}.\n\n\n\n\\section{Intersecting brane system}\n\\label{sec:setup}\n\nWe consider $D$-dimensional gravitational theory coupled to dilaton $\\phi$ and $(n_A+1)$-form gauge field ($A=1, \\ldots, m$). The action is given by\n\\begin{align}\nS=\\frac{1}{16\\pi G_D} \\int d^Dx \\sqrt{-g} \\left[ R - \\frac{1}{2} \\left( \\partial \\phi \\right)^2\n-\\sum_{A=1}^m \\frac{1}{2(n_A+2)!} e^{a_A \\phi} F_{n_A+2}^2\n\\right].\n\\end{align}\nHere $a_A$ is taken as\n\\begin{align}\na_A^2=4- \\frac{2(n_A+1)(D-n_A-3)}{D-2}\n\\end{align}\nin order to have asymptotically flat solutions, and so that extremal $q_A$-brane solutions obey a `no-force'\ncondition when the branes are static and parallel \\footnote{ \nIn this paper, we discuss the general $D$ cases.\nWhen $D=10$, all branes in type IIA and IIB superstring theory can be discussed in this setup. Then the $D<10$ cases obviously include the compactified systems of these branes in superstring theory.\nWhen $D=11$, M2- and M5-branes in M-theory can be discussed in this setup with no dilaton, $a_A=0$.\nWhen $D=12$, we expect the branes in F-theory can be discussed.\nAlthough F-theory seems a little strange in that it has two time-like directions\nand the dilaton coupling $a_A$ is pure imaginary \\cite{Khviengia:1997rh},\nif one of time-like directions is Euclideanized or all branes wind around this time-like direction, the following discussions seem to be applicable.\nHowever, at this stage, we cannot judge whether general brane systems in F-theory can be applied or not. It would be an interesting future work.}.\n\n\n\nWe compactify $(D-{d}-3)$-dimensional space as a rectangular torus $T^{D-{d}-3}$ and define its volume as $V_T$.\nTo make intersecting brane system in this setup, we put various branes on the torus.\nWe set $N_A$ $q_A$-branes \nwinding around $q_A$ cycles of the torus ($A=1,\\ldots, M$) and assume that the branes are smeared along the other cycles\\footnote{\\label{ftnt-GL}\nIf there are cycles around which no brane wind, Gregory-Laflamme transitions occur on these cycles and the black brane is localized on these cycles, when the size of the event horizon of the black brane is smaller than the size of the cycles. \nIn such cases, we cannot assume that the branes are smeared on these cycles and the effective action for the branes (\\ref{gen-effective-action}) is modified as argued in \\cite{Morita:2014ypa}.\nBy using this modified effective action, we can investigate the Gregory-Laflamme transitions in terms of the interacting separated branes. \n}. \nWhen we set the branes, we respect the intersection rule: $q_A$-brane and $q_B$-brane wind around the same $\\bar{q}$ cycles of the torus, where $\\bar{q}$ is determined as\n\\begin{align}\n\\bar{q}=\\frac{(q_A+1)(q_B+1)}{D-2}-1-\\frac{1}{2}\\epsilon_A a_A \\epsilon_B a_B\n\\label{intersection-rule} \n\\end{align}\nwhere $\\epsilon_A$ is 1 ($-1$), if $q_A$-brane is electrically (magnetically) coupled to the field strength $F_{n_A+2}$ with $q_A=n_A$ ($q_A=D-n_A-4$). \nIf all the branes satisfy this rule each other, `no-force' work among them when they are static.\nAlso we constrain ${d} \\ge 1$ so that the co-dimension of the branes is higher than two.\n(See Table \\ref{table-branes} as an example.)\n\nWe will consider the black brane solution for this setup and argue that their thermodynamical properties can be\nexplained via the microscopic theory of interacting branes (up to numerical factors).\nIt will work in general intersecting brane system as far as the branes satisfy the intersection rule.\nHowever,\ndiscussion on the general systems is rather complicated, and hence we first demonstrate it for a two brane system.\n\n\n\n\\section{Example: two brane system}\n\\label{sec:twobrane}\n\n\\begin{table}\n\\begin{eqnarray*}\n\\begin{array}{l|c|c|c|c|c|c|c|c|}\n& t & 1 & \\cdots & {d}+2 & T^{q_1-\\bar{q}} & T^{q_2-\\bar{q}} & T^{\\bar{q}} & T^{r} \\\\\n\\hline\nN_1~q_1\\text{-brane} & - & && &- && - & \\\\ \\hline\nN_2~q_2\\text{-brane}& - & && & &-& - & \\\\ \\hline\n\\end{array}\n\\end{eqnarray*}\n\\caption{The brane configuration of intersecting two brane system.\nHere $r= D- {d}-3 -(q_1+q_2-\\bar{q})$ and $\\bar{q}$ is fixed by (\\ref{intersection-rule}).\n$N_1$ $q_1$-branes wind around $q_1$ cycles of $T^{D-{d}-3}$ and are smeared on the other directions of $T^{D-{d}-3}$. \n$N_2$ $q_2$-branes wind around $q_2$ cycles and $\\bar{q}$ of them are the same cycles to the $q_1$-branes. \n}\n\\label{table-branes}\n\\end{table}\n\nWe consider the intersecting brane system which consists of $N_1$ $q_1$-branes and $N_2$ $q_2$-branes.\nThey wind around $q_1$ and $q_2$ cycles of the torus,\nsharing $\\bar{q}$ cycles so that they satisfy the intersecting rule (\\ref{intersection-rule}).\nNote that, since $\\bar{q}$ has to be a non-negative\ninteger, $q_A$ and $D$ are restricted\nto particular combinations of values.\n\nIf we focus on only the non-compact $({d}+3)$-dimensional spacetime, the $q_A$-branes behave as BPS particles with mass $m_A \\equiv \\mu_A V_A$ ($A=1,2$).\nHere $\\mu_A$ is the tension of the single $q_A$-brane and $V_A$ is the volume of the $q_A$-dimensional torus which the brane winds around.\nDue to the intersecting rule, no forces work between the branes when they are static.\nHowever, if they are moving, the interactions arise.\nOur proposal is that these interactions confine the branes\nand make them compose a bound state, and \n this bound state explains the thermodynamics of the intersecting $q_1$-$q_2$ black hole.\nTo see this, we estimate the low energy effective action of this interacting brane system.\nIf the branes are well separated, the gravitational interactions dominate and the effective action for this system has the following structure\n\\begin{align}\nS_{\\text{eff}} = \\int dt \\left(\nL_\\text{kin}+L_\\text{1-grav}+L_\\text{2-grav}+L_\\text{3-grav}+\\cdots \\right).\n\\label{moduli-D1D5}\n\\end{align}\nThe derivation of this effective action is summarized in \\ref{app:eff}.\nHere\n\\begin{align}\nL_\\text{kin}=\n\\sum_{i=1}^{N_1} \\left( \\frac{m_1}{2} \\vect{v}{2}_i+ \\frac{m_1}{8} (\\vect{v}{2}_i)^2+ \\cdots \\right)\n+\\sum_{i=1}^{N_2} \\left( \\frac{m_2}{2} \\vect{v}{2}_i+ \\frac{m_2}{8} (\\vect{v}{2}_i)^2 + \\cdots \\right)\n\\label{eff-kin}\n\\end{align}\nis the kinetic term of the branes.\n $\\vec v_i \\equiv \\partial_t \\vec r_i$ and $\\vec r_i$ are the velocity and the position of the $i$-th brane in the non-compact $({d}+2)$-dimensional space.\n We have assumed that the velocity is low ($|\\vec v_i| \\ll 1$) at low energy regime and used the non-relativistic approximation. We will justify this approximation soon.\nIn addition, we have assumed that the volume of the torus is small and the motions of the branes depend on time $t$ only\\footnote{\\label{ftnt-KK}\nThe small volume assumption is not essential in the following calculations, and\nwe apply it only to make the equations simpler.\n}.\n\n$L_\\text{1-grav}$ is the interaction which arises from a single graviton (, gauge and dilaton) exchange between two branes,\n\\begin{align}\nL_\\text{1-grav}= \n\\sum_{i\\neq j}^{N_1} \\frac{\\kappa^2_{{d}+3} m_1^2}{8{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij}}\n+\n\\sum_{i\\neq j}^{N_2} \\frac{ \\kappa^2_{{d}+3} m_2^2}{8{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij}}\n+\\sum_{i=1}^{N_1} \\sum_{j=1}^{N_2} \\frac{ \\kappa^2_{{d}+3} m_1 m_2}{{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{2}_{ij}}{\\vect{r}{{d}}_{ij}}.\n\\label{eff-2-body-D1-D5}\n\\end{align}\nHere $\\vec{r}_{ij}\\equiv \\vec{r}_i-\\vec{r}_j$ and $\\vec{v}_{ij}\\equiv \\vec{v}_i-\\vec{v}_j$ denote the relative position and the relative velocity of the $i$-th and $j$-th branes, and $\\kappa^2_{{d}+3} \\equiv \\kappa_D^2\/V_T $ is the $({d}+3)$-dimensional gravitational coupling constant.\n$\\Omega_{ d+1}\\equiv 2\\pi^{\\frac{ d}{2}+1}\/\\Gamma(\\frac{ d}{2}+1)$ is the volume of a unit $( d+1)$-sphere.\nThe first and second terms describe the two-body interactions between two $q_1$-branes and two $q_2$-branes, respectively.\nThe third term is the two-body interactions between a $q_1$-brane and a $q_2$-brane.\nThere are higher order terms of $\\vec{v}_{ij}$ but we have omitted them in this equation, since $|\\vec{v}_{ij}|$ would be small in the low energy regime.\nNote that the power of $\\vec{v}_{ij}$ of the third term is lower than the others, and\nit implies that the third term would dominate.\nHence we treat this term separately and define it as\n\\begin{align}\nL_1 \\equiv \\sum_{i=1}^{N_1} \\sum_{j=1}^{N_2} \\frac{ \\kappa^2_{{d}+3} m_1 m_2}{{d}\\Omega_{{d}+1}} \\frac{\\vect{v}{2}_{ij}}{\\vect{r}{{d}}_{ij}}.\n\\end{align}\nWe will soon see that it indeed becomes relevant.\n\nSimilarly the effective action has various interaction terms through the multi-graviton exchanges.\nHere we write down only the terms proportional to the lowest power of $v$, since they will become relevant at low energy.\nAmong 3-graviton exchange interactions, the following term will be relevant, \n\\begin{align}\nL_\\text{3-grav} \\ni L_2 \\sim\n\\sum_{i=1}^{N_1} \\sum_{j=1}^{N_1} \\sum_{k=1}^{N_2} \\sum_{l=1}^{N_2}\n\\frac{\\kappa^6_{{d}+3} m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\left( \\frac{\\vect{v}{4}_{ij}}{\\vect{r}{{d}}_{ij} \\vect{r}{{d}}_{ik} \\vect{r}{{d}}_{il} }+ \\cdots \\right).\n\\label{eff-4-body}\n\\end{align}\nWe define this term as $L_2$.\n`$\\sim$' in this article denotes equality not only including dependence on physical parameters but also including all factors of $\\pi$.\nWhen we derive these interactions in \\ref{app:eff}, we do not fix the precise numerical coefficients of these interactions.\nSince we will consider an order estimate for the thermodynamics of this interacting brane system, the precise expressions are not important. \nSimilarly the system has the following interactions\n\\begin{align}\nL_{n} \\sim & \\sum_{i_1, \\dots, i_n}^{N_1}\\sum_{j_1, \\dots, j_n}^{N_2}\n\\left( \\kappa_{{d}+3}^{2(2n-1)} \\frac{m_1^n m_2^n }{\\Omega_{{d}+1}^{2n-1}} \\prod_{k=2}^n \\prod_{l=1}^n\n\\frac{ 1 }{ \\vect{r}{{d}}_{i_1 i_k} \\vect{r}{{d}}_{i_1 j_l} } \\vect{v}{2n} + \\cdots \\right),\n\\label{moduli-multi-graviton} \n\\end{align}\nwhich describes the $2n-1$ graviton exchange among $n$ $q_1$-branes and $n$ $q_2$-branes.\n\n\nFrom now, we estimate the dynamics of this system by using the virial theorem.\nWe first assume that the branes are confined due to the interactions, and the branes satisfy\n\\begin{align}\n\\vec{v}_{ij} \\sim v, \\qquad \\vec{r}_{ij} \\sim r.\n\\label{assumption-scales}\n\\end{align}\nHere $v$ and $r$ are the characteristic scales of the velocity and size of the branes in \nthe bound state which do not depend on the species of the branes.\n(Note that since the masses of the $q_1$-brane and $q_2$-brane are generally different, we naively expect that these scales should depend on the species of the branes.\nHowever we will soon see that it does not occur in the bound state.)\nThen we can estimate the scales of the terms (\\ref{eff-kin}), (\\ref{eff-2-body-D1-D5}) and (\\ref{eff-4-body}) in the effective Lagrangian as\n\\begin{align}\nL_\\text{kin} \\sim & N_1 m_1 v^2+ N_1 m_1 v^4+ \\cdots + N_2 m_2 v^2+ N_2 m_2 v^4 + \\cdots, \\nonumber \\\\\nL_\\text{1-grav} \\sim& \\frac{ \\kappa^2_{{d}+3} N_1^2 m_1^2}{\\Omega_{{d}+1}} \\frac{v^4}{r^{{d}}}+\\frac{ \\kappa^2_{{d}+3} N_2^2 m_2^2}{\\Omega_{{d}+1}} \\frac{v^4}{r^{{d}}} + \\frac{ \\kappa^2_{{d}+3} N_1 N_2 m_1 m_2}{\\Omega_{{d}+1}} \\frac{v^2}{r^{{d}}} + \\cdots, \\nonumber \\\\\nL_\\text{3-grav} \\sim &\n\\frac{ \\kappa^6_{{d}+3} N_1^2 N_2^2 m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\frac{v^4}{r^{3{d}}} + \\cdots.\n\\label{estimate-D1D5}\n\\end{align}\nThe Lagrangian also have other terms (\\ref{moduli-multi-graviton}) but we will consider them later.\nHere we consider which terms in (\\ref{estimate-D1D5}) dominate at the low energy where $v$ would be small ($v \\ll 1 $).\nIn the second line of (\\ref{estimate-D1D5}), the third term which is from $L_1$ (\\ref{eff-2-body-D1-D5}) would dominate, since the power of $v$ is lowest \\footnote{\\label{ftnt-reduction} If the numbers and masses of the branes are quite different, e.g. $m_1 N_1 v^2 \\ll m_2 N_2 $, we can ignore the contribution from the another species of the branes and\n the system would reduce to the single-species brane system with no intersecting which has been studied in \\cite{Morita:2013wfa}. \n }.\nSuppose that this term is balanced to the term in the third line which is from $L_2$ (\\ref{eff-4-body}) due to the virial theorem, \nwe obtain the relation between $v$ and $r$ as\n\\begin{align}\n\\frac{ \\kappa^2_{{d}+3} N_1 N_2 m_1 m_2}{\\Omega_{{d}+1}} \\frac{v^2}{r^{{d}}}\n \\sim \n\\frac{ \\kappa^6_{{d}+3} N_1^2 N_2^2 m_1^2 m_2^2}{\\Omega_{{d}+1}^3} \\frac{v^4}{r^{3{d}}} \\quad \\Longrightarrow \\quad \nv^2 \\sim \\frac{\\Omega_{{d}+1}^2r^{2{d}}}{ \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2}.\n\\label{scale-v-r}\n\\end{align}\nNow we see that the other terms listed in (\\ref{estimate-D1D5}) are indeed subdominant at this scaling.\nTo see this, it is convenient to define $\\cQ_A \\equiv \\kappa^2_{{d}+3} m_A N_A\/\\Omega_{{d}+1}$.\nThen the scaling relation (\\ref{scale-v-r}) is rewritten as $v^2 \\sim r^{2 {d}}\/\\cQ_1 \\cQ_2$ and the low energy $|v| \\ll 1$ indicates $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are small\\footnote{More precisely speaking, $|v| \\ll 1$ indicates that $r^{2d}\/\\cQ_1\\cQ_2 \\ll 1$ and does not indicate that $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are both small.\nFor example, if $\\cQ_1\/r^d \\ll \\cQ_2\/r^d$, $r^d\/\\cQ_2$ can be large. However this is the situation that we should ignore another species of the brane as we argued in footnote \\ref{ftnt-reduction}, and we can exclude this possibility. Hence we can regard that $r^d\/\\cQ_1$ and $r^d\/\\cQ_2$ are both small. }.\nThus the terms in the Lagrangian (\\ref{estimate-D1D5}) scale as,\n\\begin{align}\nL_\\text{kin} \\sim & \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^d}{\\cQ_2} + \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^d}{\\cQ_2} \\frac{r^{2d}}{\\cQ_1 \\cQ_2} + \\cdots + \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^{d}}{\\cQ_1 } + \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\frac{r^{d}}{\\cQ_1 } \\frac{r^{2d}}{\\cQ_1 \\cQ_2} + \\cdots, \\nonumber \\\\\nL_\\text{1-grav} \\sim& \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\left( \\frac{r^{d}}{\\cQ_2 }\\right)^2\n+ \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} \\left( \\frac{r^{d}}{\\cQ_1 }\\right)^2+ \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} + \\cdots, \\nonumber \\\\\nL_\\text{3-grav} \\sim & \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}} + \\cdots.\n\\label{r-Q}\n\\end{align}\nHere the ordering of the terms is the same as in eq.\\,(\\ref{estimate-D1D5}).\nWe see that $L_1$ and $L_2$ scale as $ \\Omega_{{d}+1} r^{{d}}\/\\kappa^2_{{d}+3} $, while the other terms earn the factors of $r^{d}\/\\cQ_1 $ and\/or $r^{d}\/\\cQ_2 $ and are suppressed at low energy ($r^d \\ll \\cQ_1, \\cQ_2$).\nThis self-consistently ensures our assumption that $L_1$ and $L_2$ are relevant at the low energy, and hence the scaling relation (\\ref{scale-v-r}) is confirmed.\nNote that the masses of the branes always appear as the combination $m_1 m_2$ in $L_1$ and $L_2$, and it ensures that the scales of the position $r$ and velocity $v$ are independent of the species of the branes as we assumed in (\\ref{assumption-scales}).\n\nSo far we have considered the terms up to $L_2$, and now we consider $L_n$ (\\ref{moduli-multi-graviton}) ($n \\ge 3$) too. \nWe can see that at the scaling (\\ref{scale-v-r}) which was derived via the virial theorem $L_1 \\sim L_2 $, all the other interactions $L_n$ also become the same order.\nIt means that the branes are strongly coupled in the bound state.\nWe called such a bound state as `warm $p$-soup' in Ref.~\\cite{Morita:2013wfa}.\n\nFrom now, we evaluate the thermodynamical quantities of the bound state.\nBy substituting the relation (\\ref{scale-v-r}) to the Lagrangian $L \\sim L_1$, we estimate the free energy of the system as\n\\begin{align}\nF \\sim L_1 \\sim \\frac{ \\Omega_{{d}+1} r^{{d}}}{\\kappa^2_{{d}+3}}.\n\\label{free-energy-D1D5} \n\\end{align}\nHere we consider temperature dependence.\nWhen the bound state is thermalized, we treat $\\vec{r}_i$ as a thermal field (particle) and expand $\\vec{r}_i(t) = \\sum_n \\vec{r}_{i(n)} \\exp\\left(i \\frac{2 \\pi n}{\\beta}t \\right) $. \nHence we assume that the velocity $v = \\partial_t r$ are characterized by the temperature of the system through\n\\begin{align}\nv \\sim \\pi T r.\n\\label{key-assumption}\n\\end{align}\nNote that such a relation is not held generally if the system has a mass gap, but\nthere would be no mass gap in the interacting brane system as argued in Ref.~\\cite{Morita:2013wfa}.\nThen, from (\\ref{scale-v-r}), we obtain the relation between the size of the bound state and the temperature\n\\begin{align}\nr \\sim \\left( \\frac{ \\pi^2 \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2 T^2 }{\\Omega_{{d}+1}^2} \\right)^{\\frac{1}{2 ({d}-1)}} .\n \\label{scale-r-T}\n\\end{align}\nHere we have assumed ${d} \\neq 1$, and we will consider ${d}=1$ case later.\nBy substituting this relation into the free energy (\\ref{free-energy-D1D5}), \nwe estimate the entropy of the bound state as\n\\begin{align}\n&S_{\\text{entropy}} = - \\frac{\\partial F}{\\partial T} \\sim\n\\left( \\pi^2 N_1 N_2 m_1 m_2 \\right)^{\\frac{{d}}{2 ({d}-1)}}\n\\left( \\frac{ \\kappa^2_{{d}+3} T }{\\Omega_{{d}+1}} \\right)^{\\frac{1}{ {d}-1}}.\n\\label{entropy-D1D5} \n\\end{align}\n\n\nWe compare the obtained quantities with the corresponding black hole solution.\nIn the near extremal regime, the black hole thermodynamics tells us, \n\\begin{align}\nF&= -\n\\frac{d-1}{2}\\frac{\\Omega_{{d}+1} r_H^{{d}}}{\\kappa^2_{{d}+3}},\n\\label{free-energy-D1D5-GR} \\\\\nS_{\\text{entropy}}&=\n\\left( \\pi^2 N_1 N_2 m_1 m_2 \\right)^{\\frac{{d}}{2 ({d}-1)}}\n\\left( \\frac{2^{2 d+1} \\kappa^2_{{d}+3} T }{{ d}^{ d+1} \\Omega_{{d}+1}} \\right)^{\\frac{1}{ {d}-1}},\n\\label{entropy-D1D5-GR} \\\\\nr_H&= \\left( \\frac{ 64 \\pi^2 \\kappa^4_{{d}+3} N_1 N_2 m_1 m_2 T^2 }{{ d}^4\\Omega_{{d}+1}^2} \\right)^{\\frac{1}{2 ({d}-1)}} .\n\\label{eq-tmp-horizon}\n\\end{align}\nThe derivation of these expressions is summarized in \\ref{app:thermo}, and we have taken $M=2$.\nHere $r_H$ is the location of the horizon.\nTherefore, if we identify the size of the bound state $r$ with the horizon $r_H$, our result (\\ref{free-energy-D1D5}), (\\ref{scale-r-T}) and (\\ref{entropy-D1D5}) reproduce the parameter dependences of the black hole thermodynamics including $\\pi$.\n($r_H$ depends on the coordinate and we have argued what coordinate is natural in \\cite{Morita:2013wfa}.)\nThis agreement may indicate that \nthe interacting $q_1$ and $q_2$-branes described by the effective action (\\ref{moduli-D1D5}) provide\nthe microscopic origin of the $q_1$-$q_2$ black hole thermodynamics.\n\nNow we comment on the assumption $r^d \\ll \\cQ_1, \\cQ_2$ which we have used when we consider the effective action (\\ref{moduli-D1D5}).\nAt the scale (\\ref{scale-r-T}), this relation becomes\n\\ba\nT \\ll \\frac{(\\cQ_1)^{\\frac12-\\frac1d}}{\\pi (\\cQ_2)^{\\frac12}},\\quad\n\\frac{(\\cQ_2)^{\\frac12-\\frac1d}}{\\pi (\\cQ_1)^{\\frac12}}.\n\\ea\nSince we consider here the situation $\\cQ_1\/\\cQ_2\\sim \\cO(1)$, this means \n$T \\ll 1\/(\\cQ_1)^\\frac1d, 1\/(\\cQ_2)^\\frac1d$ and this is the near extremal limit in supergravity. \nThus our analysis is valid when we consider the near extremal black holes.\n\n\nFinally we consider the ${d}=1$ case.\nIn this case the relations (\\ref{scale-v-r}) and (\\ref{key-assumption}) fix $T$ as\n\\begin{align}\nT \\sim \\left( \\frac{\\Omega_{2}^2}{ \\kappa^4_{4} N_1 N_2 m_1 m_2 \\pi^2 } \\right)^{\\frac{1}{2 }} .\n\\end{align}\nand $r$ remains as a free parameter.\nThis is the Hagedorn behavior, and this temperature is coincident with the Hagedorn temperature in supergravity (\\ref{HagedornT}).\nThus the $p$-soup proposal works even in the $d=1$ case too.\n\n\n\\section{General intersecting black brane}\n\\label{sec:general}\n\nWe discuss general black brane system with arbitrary number of species of branes. \nWe can also introduce momentum along one of the cycles of the torus if all the branes wind around this cycle. (We define $R$ as the radius of this cycle.)\nIn this case, the momentum is quantized as $N_P\/R$ and there are $N_P$ gravitational waves each of which carries a momentum $1\/R$.\nWe can regard this gravitational wave as a 0-brane with mass $1\/R$ via the KK reduction of this cycle.\nAlthough the setup is complicated, we will see a simple result that the effective theory of the separated intersecting branes explains the black hole thermodynamics of the corresponding black brane at low energy.\n\nIn \\ref{app:eff}, we argue the derivation of the low energy effective action for $N_A$ $q_A$-branes ($A=1,\\ldots, M$) where $M$ is the number of the species of the charges (including momentum if we introduce it).\nThe dominant terms of the action are given by\n\\begin{align}\n\\label{lag-gen}\nS_{\\text{eff}} =& \\int dt \\left(L_1+L_2+\\cdots\\right), \\\\\nL_n \\sim &\n\\sum_{i_1,\\ldots,i_n}^{N_1}\\sum_{j_1,\\ldots,j_n}^{N_2}\\cdots \\sum_{k_1,\\ldots,k_n}^{N_M}\n\\frac{\\kappa_{d+3}^{2(nM-1)}\\,\\hat\\prod_A\\,m_A^n}{\\Omega_{d+1}^{nM-1}}\n\\left(\\vect v{2n}\\prod_{a=2}^n\\prod_{b=1}^n\\cdots\\prod_{c=1}^n\\frac{1}{\\vect rd_{i_1i_a}\\vect rd_{i_1j_b}\\cdots \\vect rd_{i_1k_c}}+\\cdots\\right),\n\\end{align}\nwhere $\\hat\\prod_A$ is the product including momentum.\nSimilar to the two brane case, we apply the virial theorem and estimate the thermodynamics of this system.\nWith the assumption (\\ref{assumption-scales}), the first two terms of the action scale as \n\\ba\nL_1 \\sim \\frac{\\kappa_{d+3}^{2(M-1)}}{\\Omega_{d+1}^{M-1}} \\left(\\hat{\\prod_A}\\, m_A N_A \\right) \\frac{v^2}{r^{d(M-1)}} \n\\,,\\quad\nL_2 \\sim \\frac{\\kappa_{d+3}^{2(2M-1)}}{\\Omega_{d+1}^{2M-1}} \\left(\\hat{\\prod_A}\\, m_A N_A \\right)^2 \\frac{v^4}{r^{d(2M-1)}} \n\\,,\n\\label{L1L2-gen}\n\\ea\nTherefore through the virial theorem $L_1\\sim L_2$, we obtain the relation between $v$ and $r$ as\n\\ba\nv^2 \\sim \\frac{\\Omega_{d+1}^M r^{dM}}{\\kappa_{d+3}^{2M}\\,\\hat\\prod_A\\, N_A m_A}.\n\\ea\nThen the free energy can be estimated as\n\\ba\nF\\sim L_1 \\sim \\frac{\\Omega_{d+1}r^d}{\\kappa_{d+3}^2}\\,.\n\\ea\nNote that this expression is common for all $M$.\nMoreover, if we assume the relation between the velocity and temperature of the system $v\\sim \\pi Tr$, we can estimate the radius of horizon\n\\ba\nr \\sim \\left(\\frac{\\pi^2 T^2 \\kappa_{d+3}^{2M} \\,\\hat\\prod_A\\, N_A m_A}{\\Omega_{d+1}^M}\\right)^{\\frac{1}{dM-2}}\\,.\n\\ea\nfor $dM-2 \\neq 0$.\nUsing the expressions of free energy and horizon radius, the entropy can be written as \n\\ba\nS_{\\text{entropy}}=-\\frac{\\partial F}{\\partial T} \n\\sim \\left(\\pi^2\\,\\hat{\\prod_A}\\, N_Am_A\\right)^{\\frac{d}{dM-2}}\\left(\\frac{\\kappa^2_{d+3}}{\\Omega_{d+1}}\\right)^{\\frac{2}{dM-2}} T^{\\frac{2d}{dM-2}-1}\\,.\n\\ea\n\nNow we compare our results with the corresponding black hole thermodynamics.\nThe results in the black hole are shown in eqs.\\,(\\ref{rH:sugra}), (\\ref{en:sugra}) and (\\ref{F:sugra}).\nWe can check that our results are coincident with the black hole thermodynamics up to rational numerical factors.\nSimilar consistency can be also seen for the Hagedorn case ($dM-2=0$).\n\n\\section{Discussions}\n\\label{sec-conclusion}\n\nWe generalized the $p$-soup proposal \\cite{Morita:2013wfa} to the intersecting brane systems.\nWe figured out that, in case the branes satisfy the intersection rule (\\ref{intersection-rule}), the bound state of the branes exhibits the thermodynamical properties which agree with the corresponding black brane.\nAlthough the intersecting brane systems are complicated themselves, this result is strikingly simple, and we believe that our proposal captures profound natures of the black hole microstates.\n\nAlso we can apply the results to superstring theory and M-theory.\nFor example, the D1-D5 brane system studied in \\cite{Morita:2014cfa} can be mapped to other brane configurations such as D0-D4 system or M5-P system via string dualities \\cite{Martinec:1999sa}.\nThen we can investigate the microstates of these branes similarly.\nFurthermore the phase transitions between these branes \\cite{Martinec:1999sa} could be understood microscopically in a fashion of \\cite{Morita:2014ypa}.\nIn such a way, we can study the various black brane dynamics in string theory through our proposal.\n\nMoreover, our discussion would be important to investigate supersymmetric gauge theories. \nIn the case of the single-species D$p$\/M-branes, through the gauge\/gravity correspondence, the duality between the black branes and the supersymmetric gauge theories on the branes at finite temperature is expected \\cite{Itzhaki:1998dd}.\nWe can explain this duality by using the fact that the effective action for the interacting $N$ branes (a variety of (\\ref{single-brane})), which is obtained from gravity, is also derived from the supersymmetric gauge theory on the branes as a low energy effective action \\cite{Morita:2013wfa}.\nNot only that, the effective action might be useful to estimate several quantities in the gauge theory which cannot be calculated in the gravity \\cite{Morita:2014ypa}.\nIn the case of the intersecting brane systems, much more varieties of supersymmetric gauge theories\nappear on the branes than the single-species brane case.\nThere, the effective action (\\ref{lag-gen}) may play a key role to understand the dynamics of these gauge theories.\n\n\nIn this way, the $p$-soup proposal may be important in the various contexts in theoretical physics. In order to establish the proposal, we have to derive the thermodynamical quantities from the effective action (\\ref{lag-gen}) exactly and compare them with the black hole results. \nHowever even fixing the coefficients of the interactions in the effective action (\\ref{lag-gen}) is difficult in the intersecting brane system, and we need to find a clever way to solve this issue.\nIf we could achieve it, it must be a valuable step toward the understanding of the black hole microstate.\n\n\n\\subsection*{Acknowledgments}\nWe would like to thank Andrew Hickling, Toby Wiseman and Benjamin Withers for useful discussions through the collaboration.\nWe also thank Nobuyoshi Ohta for useful comments.\nThe work of T.~M. is supported in part by Grant-in-Aid for Scientific Research (No.\\,15K17643) from JSPS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nIn recent years the physics community has devoted an increasing\nattention to anomalous properties of physical systems (e.g., anomalous\ntransport, anomalous diffusion, anomalous conductivity, etc.). Such\nproperties have proven \nrelevant in many fields such as thermal conductivity, kinetic\nequations, plasma physics, etc. and they are widely believed to be\ndynamical in nature. In fact, such phenomena seem to depend on the\nweak chaotic properties of the underling dynamics, see \\cite{Za} and\nreferences therein for a detailed discussion. The \nbasic idea is that, while uniformly hyperbolic dynamics gives rise to\nnormal transport properties (consider for example the diffusive\nbehavior in a finite horizon Lorentz gas \\cite{BuSi})\nnon-uniform hyperbolicity gives rise to different behavior\n(e.g. the anomalous diffusion believed to occur in infinite horizon\nLorentz gas \\cite{Bl}) due to weaker\nmixing properties (e.g. polynomial decay of correlations) and regions\nin which the motions is rather regular and where the systems spend a\nsubstantial fraction of time ({\\em sticky} regions). \n\nUnfortunately, the theoretical understanding of dynamical models with\npolynomial decay of correlations is extremely limited, hence the\nnecessity to rigorously investigate relevant toy models. The only well\nunderstood cases are expanding one dimensional maps with a neutral\nfixed point. Such maps were proposed as a model of intermittent\nbehavior in fluids (\\cite{PM}) and have been widely studied. It has been\nproven that such maps enjoy polynomial decay of correlations with the\nrate depending on the behavior of the fixed point \\cite{LSV, Yo1,\nHu1, Hu2, Sa, Go1}. In addition, when the\ndecay of correlation is sufficiently slow, the observables do not\nsatisfy the Central Limit Theorem or the Invariance Principle but\nrather, when properly rescaled, some stable law (\\cite{Zw, Go2, PoSh}).\n\nSome, more partial, results exist for multidimensional expanding\nmaps \\cite{PoYu} as well. Yet, the usual physically relevant models are\nconnected to Hamiltonian dynamics and, to our knowledge, no rigorous\nresults are available in such a situation. The simplest case which\nretain some Hamiltonian flavor is clearly a two dimensional area\npreserving map. In fact, mixing area preserving maps of the two\ndimensional torus with a neutral fixed point (the simplest type of\nsticky set) have been investigated\nnumerically \\cite{ArPr} predicting the possibility of a polynomial decay\nof correlations. \nIn this paper we consider a class of non-uniformly hyperbolic symplectic maps \\eqref{eq:map}\nof the two torus where the hyperbolicity breaks down because of a non-hyperbolic fixed point. In fact, the linearized\ndynamics at the fixed point is a shear \\eqref{eq:der}. We prove that\nthe decay of correlations is polynomial and, more precisely, decays at\nleast as $n^{-2}$ and, in some sense, one cannot expect much\nmore. Note that the example treated numerically in \\cite{ArPr} is a\nspecial case of the present setting. In \\cite{ArPr} the predicted\ndecay was $n^{-2.5}$. This emphasizes the difficulty to investigate\nsuch issues and the strong need for more theoretical work on the subject.\n\n\nThe result in the present paper is based on a precise quantitative\nanalysis of the angle between the stable and the unstable\ndirection. This angle turns out to degenerate approaching the origin\n(where the non hyperbolic fixed point is located). Once such a control\nis achieved it is possible to obtain a bound on the expansion and\ncontraction in the system. Such expansion turns out to be only\npolynomial, in contrast with the uniformly hyperbolic case where it is\nexponential. In turn, the bound on the expansion allows to study the\nregularity of the stable and unstable foliation. It turns out that\nthey are $\\Co^1$ away from the origin. This suffice to apply a simple\nrandom approximation technique that allows estimating the speed of the\ncorrelations. \n\nAs the rate of convergence to equilibrium is of order $n^{-2}$, see Theorem\n\\ref{thm:main}, the\nCentral Limit Theorem holds for zero average observable, see Corollary \n\\ref{cor:clt},\nso the model does not exhibit anomalous statistical behavior in this\nrespect. Yet, it clearly exhibits an intermittent behavior and it shows the\nmechanism whereby slow decay of correlations may arise. The present\nwork emphasizes the need to carry out similar studies in cases where the\nset producing intermittency has a more complex structure than a simple\nisolated point.\n\nThe paper is organized as follows: section \\ref{sec:results} details\nthe model and makes precise the results. Section \\ref{sec:fix} studies\nthe local dynamics at the fixed point and, in particular the\nproperties of its stable and unstable manifolds. This can be achieved\nin many way, here we find most efficient to apply a variational\ntechnique. Section \\ref{sec:narrow} establishes a precise bound for\nthe angle between the stable and the unstable direction at each\npoint. As anticipated, such a bound yields an a priory bound on the\nexpansion and contractions rates in the systems, these are obtained in\nsection \\ref{sec:expansion}. The latter result suffices to apply\nstandard distortion estimates that, in turn, allow to prove precise\nresults on the regularity of the invariant foliation and the\nholonomies, see section \\ref{sec:regularity} and section\n\\ref{sec:holo} respectively. Next, in section\n\\ref{sec:random} we introduce a random perturbation of the above map\nand investigate its statistical properties that, thanks to the added\nrandomness, can be addressed fairly easily. The relevance of the above\nrandom perturbation is that the limit of zero noise allows to easily obtain a\nbound on the rate of mixing in the original map, we do this in section\n\\ref{sec:decay}. Finally, in section \\ref{sec:lower}, we show that the\nobtained bound is close to being optimal. The paper ends with Remark\n\\ref{rem:problems} pointing to the unsatisfactory nature of some of\nthe present results and the need to investigate the related open\nproblems.\n\n\n\n\\section{The model and the results}\n\\label{sec:results}\nFor each $h\\in\\Co^{\\infty}(\\To^1,\\To^1)$ we define the map\n$T:\\To^2\\to\\To^2$ by\\footnote{\\label{foot:uno} Note that the following formula is\nequivalent, by the symplectic change of variable $q=x-y$, $p=y$, to\nthe map\n\\[\n\\widetilde T(q,p)=\\begin{cases}\n q+p&\\quad\\mod 1\\\\\n p+h(q+p)&\\quad \\mod 1\n \\end{cases}\n\\]\nwhich belongs to the standard map family. Yet, the functions $h$\nconsidered here differ substantially from the sine function which would\ncorrespond to the classical Chirikov-Taylor well known example.}\n\\begin{equation}\\label{eq:map}\nT(x,y)=\\begin{cases}\n x+h(x)+y&\\quad\\mod 1\\\\\n h(x)+y&\\quad\\mod 1\n \\end{cases}\n\\end{equation} \nWe moreover require the following properties\n\\begin{enumerate}\n\\item $h(0)=0$ (zero is a fixed point);\n\\item $h'(0)=0$ (zero is a neutral fixed point)\n\\item $h'(x)>0$ for each $x\\neq 0$ (hyperbolicity)\n\\end{enumerate}\nNote that conditions (2--3) imply that zero is a minimum for $h'$, which forces \n\\[\nh''(0)=0;\\quad h'''(0)\\geq 0.\n\\]\nWe will restrict to the generic case\n\\begin{enumerate}\n\\item[(4)] $h'''(0)>0$.\n\\end{enumerate}\nIn order to simplify the discussion we will also assume the following\nsymmetry\n\\begin{enumerate}\n\\item[(5)] $h(-x)=-h(x)$.\n\\end{enumerate}\n\nThis means that we can write\n\\begin{equation}\n\\label{eq:h}\nh(x)=bx^3+\\Or(x^5).\n\\end{equation}\n\\begin{rem}\nNote that two facts implied by the above assumptions are not necessary and could be\ndone away with at the price of more extra work: the hypothesis that\nthere is only one neutral fixed point (finitely many neutral periodic\norbits would make little difference) and the symmetry (5). We assume\nsuch facts only to simplify the presentation of the arguments.\n\\end{rem}\nSince the derivative of the map is given by\n\\begin{equation}\\label{eq:der}\nDT=\\begin{pmatrix}\n 1+h'(x)&1\\\\\n h'(x)&1\n \\end{pmatrix}\n\\end{equation}\n$\\det(DT)=1$, thus the Lebesgue measure $m$ is an invariant measure (the\nmaps are symplectic). From now on we will consider the dynamical\nsystems $T: (\\To^2,m)\\rightarrow (\\To^2,m)$. \n\nFormula \\eqref{eq:der} and property (3) imply that \nthe cone $\\Co_+=\\{v\\in\\R^2\\;|\\;Q(v):=\\langle v_1,\\,v_2\\rangle\\geq 0\\}$\nis invariant for $DT$. In additions, it is easy to check that $D_\\xi\nT^2 \\Co_+\\subset \\hbox{int }\\Co_+\\cup\\{0\\}$ for all\n$\\xi\\in\\To^2\\backslash \\{0\\}$. From\nthis and the general theory, see \\cite{LW}, follows immediately\n\n\\begin{thm}\nThe above described dynamical systems are non-uniformly hyperbolic and mixing.\n\\end{thm} \n\\begin{exmp}\nAn interesting concrete example for the above setting is given by the function \n$h(x):=x-\\sin x$.\n\\end{exmp}\nThe question remains about the rate of mixing, this is the present\ntopic. \n\\begin{rem}\nIn the following by $\\operatorname{C}$ we designate a generic\nconstant depending only on $T$. Accordingly, its value may vary from\nan occurrence to the next. In the instances when we will need a constant\nof the above type but with a fixed value we will use sub-superscripts.\n\\end{rem}\n\\begin{thm}\\label{thm:main}\nFor each $f,g\\in\\Co^{1}(\\To^2,\\R)$, $\\int f=0$, holds\ntrue\\footnote{In fact, a slightly sharper bound holds, see \\eqref{eq:sharp}.}\n\\[\n\\left|\\int f g\\circ T^n\\right|\\leq \\operatorname{C}\n\\|f\\|_{\\Co^{1}}\\|g\\|_{\\Co^{1}}n^{-2}(\\ln n)^4.\n\\]\n\\end{thm}\n\\begin{rem}\nAs in other similar cases \\cite{LSV, L, Po} the logarithmic\ncorrection is almost certainly \nan artifact of the technique of the proof. It could probably be\nremoved by using a more sophisticated (and thus more technically\ninvolved) approach. See also section \\ref{sec:lower}.\n\\end{rem}\nForm Theorem \\ref{thm:main} many facts follow, just to give an\nexample let us mention the following result that can be obtained from\nTheorem 1.2 in \\cite{L3}.\n\\begin{cor}[CLT]\\label{cor:clt}\nGiven $f\\in\\Co^1$, $\\int f=0$, the random variable\n\\[\n\\frac 1{\\sqrt n}\\sum_{i=0}^{n-1}f\\circ T^i\n\\]\nconverges in distribution to a Gaussian variable with zero mean and\nfinite variance $\\sigma$. In addition, $\\sigma=0$ iff there exists\n$\\vf\\in L^1$ such that $f=\\vf-\\vf\\circ T$.\\footnote{In particular this means\nthat the average of $f$ on each periodic orbit must be zero.}\n\\end{cor}\nThe rest of the paper is devoted to the proof of\nTheorem \\ref{thm:main} that will find its conclusion in section\n\\ref{sec:decay}. The basic \nfact needed in the proof, a fact of independent \ninterest and made quantitatively precise in Lemma \\ref{lem:distreg1}, is\nthe following. \n\\begin{thm}\\label{thm:mainlem}\nThe stable and unstable distributions are $\\Co^1$ in $\\To^2\\backslash\\{0\\}$.\n\\end{thm}\n\n\n\n\\section{The fixed point manifolds}\\label{sec:fix}\n\n\nAs usual we start by studying the local dynamics near the fixed\npoint. The first basic fact is the existence of stable and unstable\nmanifolds. This is rather standard, yet since we need some\nquantitative information we will construct them explicitly.\n\nInstead of constructing them via usual fixed point arguments it turns\nout to be faster to use a variational method.\n\n\\subsection{A variational argument}\n\nLet us consider, in a neighborhood of zero, the function\n\\begin{equation}\\label{eq:generating}\nL(x,x_1):=\\frac 12 (x-x_1)^2+G(x)\\;;\\quad G(x):=\\int_0^xh(z) dz.\n\\end{equation}\nBy setting\n\\[\n\\begin{split}\n&y:=-\\frac{\\partial L}{\\partial x}=x_1-x-h(x)\\\\\n&y_1:=\\frac{\\partial L}{\\partial x_1}=x_1-x\n\\end{split}\n\\]\nwe have $(x_1,y_1)=T(x,y)$, that is {\\sl $L$ is a generating function\nfor the map \\eqref{eq:map}}.\n\nThen, for each $a\\in\\R$, we define the Lagrangian $\\Lp_a:\\ell^2(\\N)\\to\\R$ by\n\\begin{equation}\\label{eq:lagrange}\n\\Lp_a(x):=\\sum_{n=1}^\\infty L(x_n,x_{n+1})+L(a,x_1).\n\\end{equation}\nThe justification of the above definition rests in the following\nLemma.\n\\begin{lem}\\label{lem:critical}\nFor each $a\\in(-1,1)$, holds true $\\Lp_a\\in\\Co^1(\\ell^2(\\N))$. In addition, if\n$x\\in \\ell^2(\\N)$ is such that $D_x\\Lp_a=0$, then setting $x_0=a$ and \n$y_n:=x_{n+1}-x_n-h(x_n)$, we have $T^n(x_0,y_0)=(x_n,y_n)$.\n\\end{lem}\n\\begin{proof}\nFirst of all \\eqref{eq:h} implies that there exists $\\operatorname{C}>0$ such that\n$|G(x)|\\leq \\operatorname{C} x^4$. It is then easy to see that $\\Lp_a$ is well defined\nfor each sequence in $\\ell^2(\\N)$.\n\nNext, for each $n\\in\\N$ let us define\n$(\\nabla\\Lp_a)_n:=\\partial_{x_n}\\Lp_a$.\nClearly,\n\\[\n\\begin{split}\n(\\nabla\\Lp_a)_1(x)&=2x_1-x_2+h(x_1)-a\\\\\n(\\nabla\\Lp_a)_n(x)&=2x_n-x_{n+1}-x_{n-1}+h(x_n).\n\\end{split}\n\\]\nOf course, for $x\\in\\ell^2(\\N)$, $ \\nabla\\Lp_a(x)\\in\\ell^2(\\N)$, it is\nthen trivial to check \nthat $D_x\\Lp_a(v)=\\langle \\nabla\\Lp_a(x), v\\rangle$.\nThe last statement follows by a direct computation.\n\\end{proof}\n\nBy the above Lemma it is clear that one can obtain the stable\nmanifolds of the fixed point from the critical points of $\\Lp_a$, it\nremains to prove that such critical points do exist. We will start by\nconsidering the case $a\\geq 0$.\n\nDefine\n\\begin{equation}\\label{eq:fixedset}\nQ_B:=\\{x\\in\\ell^2(\\N)\\;|\\;\\,|x_n-\\frac A{n+c}|\\leq B(n+c)^{-\\frac\n32}\\}\n\\end{equation}\nwhere $A:=\\sqrt{\\frac 2b}$; $c:=\\frac Aa$.\n\nIt is immediate to check that $Q_B$ is compact and convex. In\naddition, if $a$ is sufficiently small, then $G$ is strictly convex on\n$[-2a,2a]$ which implies that $\\Lp_a|_{Q_B}$ is strictly\nconvex. Accordingly, $\\Lp_a$ has minimum in $Q_B$, moreover the strict\nconvexity implies that such a minimum is unique, for $a$ fixed.\n\nLet us call $x(a)$ the point in $Q_B$ where $\\Lp_a$ attains its\nminimum.\n\\begin{lem}\\label{lem:stab}\nFor $a$ small enough, $D_{x(a)}\\Lp_a=0$.\n\\end{lem}\n\\begin{proof}\nSuppose that $\\partial_{x_n}\\Lp_a(x(a))\\neq 0$ for some $n\\in\\N$, for\nexample suppose it is negative. Then $x(a)$ is on the border of $Q_B$,\nsay $x(a)_n=A(n+c)^{-1}+B(n+c)^{-\\frac 32}$, otherwise we could increase\n$x(a)_n$ and decrease $\\Lp_a$ still remaining in $Q_B$, contrary to\nthe assumption. But then\n\\[\n\\begin{split}\n\\partial_{x_n}\\Lp_a(x(a))&=2x(a)_n-x(a)_{n-1}-x(a)_{n+1}+h(x(a)_{n})\\\\\n&=\\frac {2A}{n+c}+\\frac{2B}{(n+c)^{\\frac\n32}}-x(a)_{n-1}-x(a)_{n+1}+h(\\frac {A}{n+c}+\\frac{B}{(n+c)^{\\frac\n32}}) \\\\\n&\\geq -\\frac{2A}{[(n+c)^2-1](n+c)}+\\frac\n{2A}{(n+c)^3}+\\frac{3bA^2B}{(n+c)^\\frac 72}+\\Or((n+c)^{-4})\\\\\n&+\\frac{B(n+c)^3\\{2(1-(n+c)^{-2})^{\\frac 32}-(1-(n+c)^{-1})^{\\frac\n32} -(1+(n+c)^{-1})^{\\frac 32}}{(n+c)^{\\frac 32}[(n+c)^2-1]^{\\frac\n32}}\\\\\n&=\\frac{(6-\\frac {15}4)B}{(n+c)^{\\frac 72}}+\\Or((n+c)^{-4})\\geq 0\n\\end{split}\n\\]\nprovided $a$ is sufficiently small. We have thus a contradiction. The\nother possibilities are analyzed similarly.\n\\end{proof}\n\nTo conclude we need some information on the regularity of $x(a)$ as a\nfunction of $a$. Unfortunately, the\nimplicit function theorem does not applies since $D^2\\Lp_a$ does not\nhave a spectral gap, yet for our purposes a simple estimate\nsuffices.\n\\begin{lem}\\label{lem:lipman}\n$x_1(a)$ is a Lipschitz function of $a$. Moreover, when derivable,\n\\[\n|y_0(a)'|\\leq \\operatorname{C} |x(a)_0|.\n\\]\n\\end{lem}\n\\begin{proof}\nBy Lemma \\ref{lem:stab} it follows, for each $a,a'$ sufficiently small\n\\[\n\\partial_{x_n}\\Lp_{a'}(x(a'))=\\partial_{x_n}\\Lp_{a}(x(a))=0\n\\]\nthat is\n\\[\n\\partial_{x_n}\\Lp_{a'}(x(a'))-\\partial_{x_n}\\Lp_{a'}(x(a))=\\partial_{x_n}\\Lp_{a}(x(a))\n-\\partial_{x_n}\\Lp_{a'}(x(a))\n\\]\nwhich yields\n\\begin{equation}\\label{eq:strange}\n\\begin{split}\n&(2+h'(\\xi_1))\\zeta_1-\\zeta_2=a'-a\\\\\n&(2+h'(\\xi_n))\\zeta_n-\\zeta_{n+1}-\\zeta_{n-1}=0,\n\\end{split}\n\\end{equation}\nwhere $\\zeta_n:=x(a')_n-x(a)_n$ and $\\xi_n\\in[x(a)_n,x(a')_n]$.\n\nNotice that, if $|\\zeta_n|\\geq|\\zeta_{n-1}|$, then\n\\[\n|\\zeta_{n+1}|=|(2+h'(\\xi_n))\\zeta_n-\\zeta_{n-1}|\\geq\n2|\\zeta_n|-|\\zeta_{n-1}|\\geq |\\zeta_n|.\n\\]\n\nThus, by induction, if $|\\zeta_n|\\geq |\\zeta_{n-1}|$, then $|\\zeta_m|\\geq\n|\\zeta_{n-1}|$ for each $m\\geq n$, which would imply $\\zeta_{n-1}=0$ since\n$\\zeta\\in\\ell^2(\\N)$. But then\n$(2+h'(\\xi_n))\\zeta_n=\\zeta_{n+1}$, that is\n$|\\zeta_{n+1}|\\geq|\\zeta_n|$. Accordingly, again by induction,\n$\\zeta_m=0$ for each $m\\geq n-1$. This means that we can restrict\nourselves to the case $\\zeta_n\\neq 0$, $|\\zeta_n|\\geq |\\zeta_{n+1}|$.\nHence,\n\\[\n|a'-a|=|(2+h'(\\xi_1))\\zeta_1-\\zeta_2|\\geq 2|\\zeta_1|-|\\zeta_2|\\geq\n|\\zeta_1|.\n\\]\nThat is $|x(a')_n-x(a)_n|\\leq |x_1(a')-x_1(a)|\\leq |a'-a|$.\n\nFinally, summing \\eqref{eq:strange} over $n\\in\\N$, \n\\[\n\\sum_{n=1}^\\infty h'(\\xi_n)\\zeta_n=-\\zeta_1+a'-a.\n\\]\nThus, where all the $x(a)_n$ are differentiable (a full measure set),\n$|x(a)_n'|\\leq|x(a)_1'|\\leq 1$ and\n\\begin{equation}\\label{eq:derivxy}\n\\begin{split}\nx(a)_1'&=1-\\sum_{n=1}^\\infty h'(x(a)_n)x(a)_n'\\\\\ny(a)_0'&=-\\sum_{n=0}^\\infty h'(x_n)x(a)_n'.\n\\end{split}\n\\end{equation}\nAccordingly,\n\\[\n|y_0(a)'|\\leq 6b\\sum_{n=0}^\\infty x_n^2\\leq \\operatorname{C} a.\n\\]\n\\end{proof}\n\nClearly, the above Lemma implies that, calling $(x,\\gamma_s(x))$ the\ngraph of the stable manifold, $\\gamma_s\\in Lip(-1,1)$. The case $a\\leq\n0$ and the unstable manifolds can be treated similarly, yet there\nexists a faster--and more\ninstructive--way.\n\n\\subsection{Reversibility}\n\nNotice that the map $T$ is {\\sl reversible}\nwith respect to the transformations\\footnote{While the reversibility for\n$\\Pi$ is a general fact, the one for $\\Pi_1$ depends on the simplifying\nsymmetry hypothesis (5).}\n\\begin{equation}\\label{eq:invol}\n\\Pi(x,y):=(x, -y-h(x)); \\quad \\Pi_1(x,y):=(-x, y+h(x))\n\\end{equation}\nIndeed, $\\Pi^2=\\Pi_1^2=\\Id$ and $\\Pi T\\Pi=\\Pi_1 T\\Pi_1=T^{-1}$. \n\n\\begin{rem}\\label{rem:rev}\nThe reversibility implies that, for $x\\geq 0$,\n$(x,\\gamma_u(x))=\\Pi(x,\\gamma_s(x))$, and, for $x\\leq 0$,\n$(x,\\gamma_u(x))=\\Pi_1(-x,\\gamma_s(-x))$ is the unstable manifold of\nzero. \n\\end{rem}\n\n\\subsection{A quasi-Hamiltonian}\n\nTo study the motion near the fixed point it is helpful to find a local\n``Hamiltonian'' function. By Hamiltonian function we mean a function\nthat is locally invariant for the dynamics. Such a function can be\ncomputed as a formal power series starting by the relation $H\\circ\nT=H$. In fact, we are interested only in a suitable approximation. A\ndirect computation yields that, by defining $G(x):=\\int_0^x h(z)dz$ and\n\\begin{equation}\\label{eq:hamiltonian}\nH(x,y):=\\frac 12 y^2-G(x)+\\frac 12 h(x)y-\\frac 1{12}h'(x)y^2+\\frac 1 {12}h(x)^2,\n\\end{equation}\nholds true\\footnote{In fact, setting $(x_1,y_1):=T(x,y)$, holds\n\\[\n\\begin{split}\n&H(x_1,y_1)-H(x,y)=yh(x)+\\frac 12 h(x)^2-h(x)y-h(x)^2-\\frac\n12h'(x)y_1^2-\\frac 16 h''(x)y_1^3+\\frac 12h'(x)y_1^2\\\\\n&+\\frac 12 h(x)^2\n+\\frac 14 h''(x)y_1^3-\\frac 1{12}h''(x)y_1^3-\\frac\n16h'(x)h(x)y_1+\\frac 16 h'(x)h(x)y_1+\\Or(x^8+x^4y^2+y^4).\n\\end{split}\n\\]\n}\n\\begin{equation}\\label{eq:appham}\nH(T(x,y))-H(x,y)=\\Or (x^8+y^4).\n\\end{equation}\nThis approximate conservation law suffices to obtain rather precise\ninformation on the near fixed point dynamics.\\footnote{The reader\n should be aware that it is possible to do much better, that is to\n obtain an exponentially precise conservation law, see \\cite{L2},\n \\cite{BG}.} \nThe first application \nis given by the following information on the stable manifold.\n\\begin{lem}\\label{lem:manifzero}\nFor $x\\geq 0$ sufficiently small holds\n\\[\n\\gamma_s(x)= -A^{-1}x^2+\\Or(x^3).\n\\] \n\\end{lem}\n\\begin{proof}\nUsing the notation of Lemma \\ref{lem:stab}, for fixed $a$ we get\n\\begin{equation}\\label{eq:unoy}\ny_n=x_{n+1}-x_n-h(x_n)=\\Or((n+c)^{-\\frac 32}).\n\\end{equation}\nHence\n\\[\nH(x_n,y_n)=\\Or((n+c)^{-3}).\n\\]\nUsing equation \\eqref{eq:appham} we have\n\\[\nH(x_0,y_0)=H(x_n,y_n)+\\Or(\\sum_{i=0}^{n-1}x_i^8+y_i^4)\n\\]\nthat is\n\\[\n|H(x_0,y_0)|\\leq \\operatorname{C} \\left\\{(n+c)^{-3}+\n\\sum_{i=0}^{\\infty}(n+c)^{-6}\\right\\}\\leq \\operatorname{C}\n\\left\\{(n+c)^{-3}+c^{-5}\\right\\}.\n\\]\nTaking the limit for $n$ to infinity in the above expression and\nremembering the definition of $c$ follows\n\\[\n|H(x_0,y_0)|\\leq\\operatorname{C} x_0^{5}.\n\\]\nSince equation \\eqref{eq:unoy} implies $|y_0|=\\Or(x_0^{\\frac 32})$,\nfrom \\eqref{eq:hamiltonian} we have\n\\[\ny_0^2-\\frac b2 x_0^4+bx_0^3y_0=\\Or(x_0^5)\n\\]\nfrom which the lemma follows.\n\\end{proof}\nAccording to Lemma \\ref{lem:manifzero}, the local picture of the\nmanifolds is given by Figure \\ref{fig:man}.\n\\begin{figure}[ht]\\\n\\centering\n\\put(-40,0){\\line(1,0){80}}\n\\put(0,-20){\\line(0,1){40}}\n\\put(-6.5,-15){{\\tt Fat sector}}\n\\put(30,2){{\\tt Thin sector}}\n\\thicklines\n\\qbezier(-40, 10)(0,-10)(40,10)\n\\qbezier(-40, -10)(0,10)(40,-10)\n\\put(-39,10){$W^s$}\n\\put(-39,-14){$W^u$}\n\\put(40,11){$W^u$}\n\\put(40,-9){$W^s$}\n\\put(0,-24){\\ }\n\\caption{The manifolds of the fixed point}\\label{fig:man}\n\\end{figure}\n\n\\subsection{Manifold regularity}\nSince in the previous section we have seen that the manifold are\nLipschitz curves, we can define the dynamics restricted to the unstable\nmanifold:\n\\begin{equation}\\label{eq:fu}\nf_u(x):=x+h(x)+\\gamma_u(x).\n\\end{equation}\n\nOur next task is to obtain sharper information on the manifolds\nregularity.\\footnote{Of course, the manifold should be as smooth as $h$, but\nthis results is not needed in the following while we do need an\nexplicit bound on the curvature.}\n\\begin{lem}\\label{lem:manifoldreg}\nThe unstable manifold of the fixed point is $\\Co^2$, apart from zero.\n\\end{lem}\n\\begin{proof}\nIt is clearly enough to show that $\\gamma_u\\in \\Co^2$ apart from\nzero. To do so call $u(x)=\\gamma_u'(x)$ (the derivative exists almost\neverywhere since $\\gamma_u$ is Lipschitz). The tangent vector to the\nunstable manifold has the form $(1,u)$. On the other hand\n\\[\n\\begin{split}\n\\lambda_u(f_u^{-1}(x))\n\\begin{pmatrix}1\\\\u(x)\\end{pmatrix}&:=\\begin{pmatrix}\n 1+h'(f_u^{-1}(x))&1\\\\\n h'(f_u^{-1}(x))&1\n \\end{pmatrix}\n \\begin{pmatrix}1\\\\u(f_u^{-1}(x))\\end{pmatrix}\\\\\n&=:\\lambda_u(f_u^{-1}(x))\\begin{pmatrix}1\\\\F(f_u^{-1}(x),u(f_u^{-1}(x)))\n\\end{pmatrix}\n\\end{split}\n\\]\nwhere\n\\begin{equation}\\label{eq:unsdef}\n\\begin{split}\n&\\lambda_u(x):=1+h'(x)+u(x)=f'_u(x)\\\\\n&F(x,u):=1-\\frac{1}{1+h'(x)+u}.\n\\end{split}\n\\end{equation}\nAccordingly, setting $x_i:=f_u^{-i}(x)$, holds\n$u(x_i)=F(x_{i+1},u(x_{i+1}))$. Next, let $v_i:=2A^{-1}x_i$, then a\ndirect computation yields $F(x_i,v_i)-v_{i-1}=\\Or(x_i^3)$, thus\n\\[\n|u(x_i)-v_i|\\leq |F(x_{i+1},u(x_{i+1}))-F(x_{i+1}, v_{i+1})|+\\operatorname{C} x_i^3\n\\leq |u(x_{i+1})- v_{i+1}|+\\operatorname{C} x_i^3.\n\\]\nBy induction, and equation \\eqref{eq:fixedset}, it follows\n$|u(x)-2A^{-1}x|\\leq \\operatorname{C}\\sum_{i=0}^\\infty x_i^3\\leq \\operatorname{C} x^2$.\n\nOn the other hand, given a different\npoint $z$, it holds\n\\[\n\\begin{split}\nu(x)-u(z)&=F(x_1,u(x_1))-F(z_1,u(z_1))\\\\\n&=\\lambda_u(x_1)^{-1}\\lambda_u(z_1)^{-1}\n\\left\\{(u(x_1)-u(z_1))+h'(x_1)-h'(z_1))\\right\\}.\n\\end{split}\n\\]\nIterating the above equation yields\n\\begin{equation}\\label{eq:unstreg}\n\\begin{split}\nu(x)-u(z)=&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}(u(x_n)-u(z_n))\\\\\n&\\ \\ +\\sum_{k=1}^{n}\\lambda_{u,k}(x)^{-1}\\lambda_{u,k}(z)^{-1}(h'(x_k)-h'(z_k))\n\\end{split}\n\\end{equation}\nwhere $\\lambda_{u,n}(x):=\\prod_{k=1}^{n}\\lambda_u(f_u^{-k}(x))$.\nNext, let $x=a$, then, accordingly to Lemma\n\\ref{lem:stab}, equation \\eqref{eq:fixedset} and Remark \\ref{rem:rev}, we have\n$x_i= A(i+c)^{-1}+\\Or((i+c)^{3\/2})$. This means that, in a\nsufficiently small neighborhood of zero, and for $z$ sufficiently\nclose to $x$ holds\n\\[\n\\begin{split}\n\\lambda_{u,n}(x)&\\geq\n\\prod_{k=1}^{n}\\left(1+v_k-\\operatorname{C} x_i^2\\right)\n\\geq \\prod_{k=1}^{n}\\left(1+\\frac 2{k+c}-\\operatorname{C} (k+c)^{-3\/2}\\right)\\\\\n&\\geq e^{\\sum_{k=1}^{n}\\frac{2}{k+c}-\\operatorname{C} (k+c)^{-3\/2}}\n\\geq \\operatorname{C} (A^{-1}an+1)^{2}\n\\end{split}\n\\]\nprovided $x$ is close \nenough to zero. The same estimate holds for $\\lambda_{u,n}(z)$. \n\nThis implies that $u$ is continuous. Indeed, for each $\\ve>0$ choose\n$n_0(x)\\in\\N$ such that $\\lambda_{u,n_0(x)}(x_n)^{-1}\\leq \\ve$, then\n\\[\n|u(x)-u(z)|\\leq \\sum_{k=1}^{n_0(x)}|h'(x_k)-h'(z_k)|+\\ve \n\\]\nand we can thus choose $z$ close enough to $x$ such that\n$|u(x)-u(z)|\\leq 2\\ve$. Note that this implies the continuity of the\n$\\lambda_{u,n}$ as well.\n\nTo conclude, we choose $n(z)$ such that $\\operatorname{C}(A^{-1}an(z)+1)^{-4}\\geq\n|x-z|^{1+\\alpha}$, for some $\\alpha>0$, \naccordingly \n\\[\nu(x)-u(z)=\\sum_{k=1}^{n(z)}\\lambda_{u,k}(x)^{-1}\n\\lambda_{u,k}(z)^{-1}(h'(x_k)-h'(z_k))+{\\mathcal\nO}(|x-z|^{1+\\alpha})\n\\]\nSince the series is uniformly convergent we have\n\\begin{equation}\\label{eq:unstder}\nu'(x)=\\sum_{k=1}^{\\infty}\\lambda_{u,k}(x)^{-3}h''(x_k)\n\\end{equation}\nfrom which the lemma follows.\\footnote{Remark that to obtain the \nresult on a larger neighborhood it suffices to iterate the unstable \nmanifold forward.}\n\\end{proof}\n\n\n\\begin{rem}\nAll the above results for the unstable manifold $\\gamma_u$ have the\nobvious counterpart for the stable manifold $\\gamma_s$ that can be\nreadily obtained via reversibility, see Remark \\ref{rem:rev}.\n\\end{rem} \n\n\n\\section{A narrower cone field}\\label{sec:narrow}\n \nHere our goal is to estimate the angle between stable and unstable manifolds.\n\n\nMore precisely, we wish to prove that there exists two constants $K_+,K_-\n\\in\\R^+$ such that the cone field\n$\\Co_*(\\xi):=\\{(1,u)\\in\\R^2\\;|\\;K_-(|x|+\\sqrt{|y|})\\leq u\\leq\nK_+(|x|+\\sqrt{|y|})\\}$ contains the unstable direction (by\nreversibility we can also define \nthe stable cone field $\\Co^-_*$). \n\n\\begin{prop}\\label{prop:dista}\nFor each $\\xi\\in\\To^2$ holds true $E^u(\\xi)\\in\\Co_*(\\xi)$,\n$E^s(\\xi)\\in\\Co_*^-(\\xi)$.\n\\end{prop}\n\nThe rest of the section is devoted to the proof of Proposition\n\\ref{prop:dista}.\n\nClearly a problem arises only in a neighborhood of zero. Accordingly\nthe first step is to gain a better understanding of the dynamics near\nzero. \n\n\n\\subsection{Near fixed point dynamics}\\label{sub:nearfix}\nFor each $\\delta>0$ let $Q_\\delta:=[-\\delta,\\delta]^2$ be a square\nneighborhood of zero. The manifolds of the fixed point divide\n$Q_\\delta$ into four sectors: two thin and two fat (see Figure\n\\ref{fig:man}). We will discuss explicitly the dynamics in the two\nsectors below the unstable manifold (the other two being identical by\nsymmetry).\n\n\\begin{lem}\\label{lem:zdyn1}\nFor each $(x,y)\\in\\To^2$, $y\\leq \\gamma_u(x)$, let\n$(x_n,y_n):=T^n(x,y)$, then it holds true \n\\[\nx_n\\leq f_u^n(x)\\quad \\forall n\\geq 0.\n\\]\n\\end{lem}\n\\begin{proof}\nFirst note that the trajectory will always remain below the unstable\nmanifold. Hence, by induction,\n\\[\n\\begin{split}\nx_{n+1}&=x_n+h(x_n)+y_n\\leq x_n+h(x_n)+\\gamma_u(x_n)\\\\\n&\\leq f_u^n(x)+\nh(f_u^n(x))+\\gamma_u( f_u^n(x))=f_u^{n+1}(x). \n\\end{split}\n\\]\n\\end{proof}\nThe above lemma will suffice to control the dynamics in the thin\nsector, more work is needed for the fat one. In fact, when the\ntrajectories are close to the stable or the unstable manifolds the\nabove result can still be used (possibly remembering reversibility). On the\nother hand when the trajectory is close enough to zero its behavior\nis drastically different from the one on the invariant manifolds.\n\nTo define more precisely the meaning of ``close to zero'' let us\nintroduce the parabolic sector $P_M:=\\{(x,y)\\in Q_1\\;|\\;|y|\\geq Mx^2\\}$.\nWe consider a backward trajectory starting from $x\\leq 0$, $y\\leq\n\\gamma_u(x)$, the other possibilities follow by reversibility. Let, as usual, $(x_{n},\ny_n):=T^n(x,y)$, $n\\in\\Z$. Let $m_+$ be the smallest integer for which\n$(x_{-n},y_{-n})\\in P_M$, $m$ the largest integer such that\n$x_{-m}\\leq 0$, and $m_-$ the largest integer for which\n$(x_{-n},y_{-n})\\in P_M$. Define, see \\eqref{eq:hamiltonian}, \n\\[\nE:=H(x_{-m},y_{-m}).\n\\]\nIn addition, define the function $\\Upsilon_E: [-1,1]\\to\\R^-$,\nby\\footnote{Computing for $y\\leq0$ yields\n\\[\n\\begin{split}\n\\Upsilon_E(x)&=-\\frac {h(x)}2-\\sqrt{\\frac{h(x)^2}4+2[G(x)+E-\\frac\n1{12}h(x)^2](1-\\frac 16 h'(x))}\\\\\n&=-\\sqrt{2(E+G(x))}(1+\\Or(x^2))+\\Or(x^3).\n\\end{split}\n\\]\n}\n\\begin{equation}\\label{eq:upsi}\nH(x,\\Upsilon_E(x))=E.\n\\end{equation}\nThen, by \\eqref{eq:appham}, and since $|y_{-n}|\\geq \\operatorname{C} x_{-n}^2$,\n\\[\n\\begin{split}\nH(x_{-n},y_{-n})-&H(x_{-n},\\Upsilon_E(x_{-n}))=H(x_{-n},y_{-n})-H(x_{-m},y_{-m})\\leq\n\\operatorname{C} \\sum_{k=n}^m y_{-k}^4\\\\\n&\\leq\\operatorname{C} \\sum_{k=n}^m |y_{-k}^3|(x_{-k-1}-x_{-k})\n\\leq \\operatorname{C}|y_{-n}^3| |x_{-n}|. \n\\end{split}\n\\]\nAccordingly, for $n\\leq m$ it holds true\n\\begin{equation}\\label{eq:dyny}\n|y_{-n}-\\Upsilon_E(x_{-n})|\\leq \\operatorname{C}|y_{-n}|^2|x_{-n}|.\n\\end{equation}\n\\begin{lem}\\label{lem:zdyn2}\nIn the above described situation, setting $M=\\sqrt b$, the following holds true\n\\begin{enumerate}\n\\item $f_u^{-n}(x)\\leq x_{-n}\\quad \\forall\\; n\\leq m$;\n\\item $f_s^{k}(x_{-n})\\geq x_{-n+k}\\quad \\forall\\; n\\geq m_-,\\ k\\leq\n n-m_-$;\n\\item $\\sqrt{E}\\leq |y_{-n}|\\leq 3\\sqrt{E}$ for all\n $n\\in\\{m_+,\\dots, m_-\\}$;\n\\item $m_+\\leq 2A(Eb^{-1})^{-\\frac 14}$;\n\\item $ 2(12Eb)^{-\\frac 14}\\leq m_--m_+\\leq 4(Eb)^{-\\frac 14}$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe first fact is proven as in Lemma \\ref{lem:zdyn1}, the second\nfollows by reversibility. Hence, by\nthe results of section \\ref{sec:fix}, for $n\\leq m$, it follows \n\\begin{equation}\\label{eq:nest}\n|x_{-n}|\\leq |f_u^{-n}(x)|\\leq \\frac{2A|x|}{|x|n+A}.\n\\end{equation}\nNext we want to determine the points $x_{m_+}$ and $x_{m_-}$. The idea\nis to use $\\eqref{eq:dyny}$ that determines with good precision the\ngeometry of the trajectories. Let $\\bar x$ be defined by\n$\\Upsilon_E(\\bar x)=-M\\bar x^2$. Then\n\\[\n\\bar x=-\\left[\\frac{2E}{M^2-\\frac b2}\\right]^{\\frac 14}+\\Or(E^{\\frac\n 54}).\n\\] \nOn the other hand, since by definition $|y_{-m_+}|\\geq Mx_{-m_+}^2$\nand $|y_{-m_++1}|\\leq Mx_{-m_++1}^2$, holds\n$|y_{-m_+}-Mx_{-m_+}^2|\\leq \\operatorname{C}|x_{-m_+}^3|$.\nHence, by \\eqref{eq:dyny} it follows \n\\[\n|x_{-m_+}-\\bar x|\\leq \\operatorname{C}\\frac{x_{-m_+}^3}{\\min\\{\\bar\nx,x_{-m_+}\\}}\\leq \\operatorname{C} \\frac{x_{-m_+}^3}{x_{-m_+}-|x_{-m_+}-\\bar x|}.\n\\]\nSolving the above inequality yields\n\\[\n|x_{-m_+}-\\bar x|\\leq\\operatorname{C} x_{-m_+}^2\\leq \\operatorname{C}\\bar x^2\\leq\\operatorname{C}\n\\sqrt E.\n\\]\nAnalogously $|x_{-m_-}+\\bar x|\\leq \\operatorname{C} \\sqrt E$.\n\\relax From this (3) and (4) easily follows. Finally,\n\\[\n2|\\bar x|\\geq |x_{m_-}-x_{m_+}|=|\\sum_{n=m_+}^{m_-}y_n|\\geq\n\\operatorname{C}(m_+-m_-)\\sqrt E.\n\\]\nWhich implies (5).\n\\end{proof}\n\nWe are now ready to refine our knowledge of the stable and unstable\ndirection. Let us fix $\\varrho\\in(0,1\/2)$.\n\n\\subsection{The cone field--Outside $Q_\\varrho$}\n\nThe general idea is to take the positive cone field $\\Co_+$ (which is\ninvariant and contains the unstable direction) and to push it forward\nin order to obtain a narrower cone \nfield. First of all outside $Q_{\\sqrt\\varrho}$ we have (see \\eqref{eq:unsdef})\n\\[\n1\\geq F(x,u)\\geq F(x,0)\\geq 2b \\varrho,\n\\]\nwhere we have chosen $\\varrho$ small enough.\nHence the cone field $\\Co_0=\\{(1,u)\\;|\\;1\\geq u\\geq 2b\\varrho\\}$ is invariant\noutside $Q_{\\sqrt\\varrho}$. It remains to understand what\nhappens in a neighborhood of the origin of order $\\sqrt\\varrho$. \n\nLet us define $\\bar u(\\xi)$ by the equation $u=F(\\xi,u)$. An easy\ncomputation shows\n\\[\n\\bar u(\\xi)= \\frac{-h'(x)+\\sqrt{h'(x)^2+4h'(x)}}2= \\sqrt{3b}|x|+\\Or(x^2).\n\\]\nBy reversibility we can restrict ourselves to the case $x\\geq 0$, in this\ncase the only possibility to enter the region $Q_{\\sqrt\\varrho}$ is\nvia the fourth quadrant. \n\nNote that $F(\\xi,u)\\geq u$ provided $0\\leq u\\leq \\bar u(\\xi)$.\nThis means that if $\\xi\\not\\in Q_{\\sqrt\\varrho}$ but $T\\xi\\in\nQ_{\\sqrt\\varrho}$, then the lower bound of the cone $D_\\xi T^n\\Co_0$\ndoes not decreases until $\\sqrt{3b}x_i\\leq 2b\\varrho$, where\n$(x_i,y_i):=T^i\\xi$. \n\nAccordingly, the cone field $\\Co_0$ is invariant\nalso in the fourth quadrant, outside the set $Q_{2\\sqrt{\\frac b3}\\varrho}$.\nNow consider the cone field $\\Co_1(\\xi):=\\{(1,u)\\;|\\;2b\\varrho\\leq\nu\\leq L \\bar u(\\xi)\\}$, $L=(3b\\varrho)^{-\\frac 12}$, for $\\xi\\in\nQ_{\\sqrt\\varrho}\\backslash Q_{2\\sqrt{\\frac b3}\\varrho}$. Clearly, if\n$|x|\\geq \\sqrt \\varrho$, then $L\\bar u(\\xi)\\geq 1$. Hence, as the\npoint enters $Q_{\\sqrt\\varrho}$, the image of $\\Co_0$ is contained in\n$\\Co_1$, moreover we have already seen that the lower bound is\ninvariant provided $\\xi_i\\not\\in Q_{2\\sqrt{\\frac b3}\\varrho}$. Let us follow the\nupper edge, if $u\\leq L\\bar u(\\xi_i)$, then\\footnote{Note that this\ncomputation holds for all $\\xi_i\\in Q_{\\sqrt{\\varrho}}\\backslash P_M$.}\n\\[\n\\begin{split}\nF&(\\xi_i,u)=F(\\xi_i,u)-F(\\xi_i,\\bar u(\\xi_i))+\\bar u(\\xi_i)\\\\\n&\\leq \\frac{(L-1)\\bar u(\\xi_i)}{(1+h'(x_i)+L\\bar u(\\xi_i))(1+h'(x_i)+\\bar\nu(\\xi_i))}+\\bar u(\\xi_i)\\\\\n&\\leq L\\bar u(\\xi_i)-(L^2-1)\\bar u(\\xi_i)^2+\\Or(x^3_i)\\\\\n&\\leq L\\bar u(\\xi_{i+1})+\\sqrt{3b}|y_{i+1}|-(L^2-1)\\bar u(\\xi_i)^2+\\Or(x^3_i)\\\\\n&\\leq L\\bar u(\\xi_{i+1}),\n\\end{split}\n\\]\nprovided $\\xi_{i+1}\\not\\in P_M$, with $M\\leq \\sqrt{3b}(L^2-1)$, which is\nfine provided $\\varrho$ is chosen small enough. The above discussion\ncan be summarized as follows.\n\\begin{lem}\n\\label{lem:conefield-one}\nThere exists $\\varrho>0$: For $\\xi\\not\\in Q_{2\\sqrt{\\frac b3}\\varrho}$\nthe unstable distribution is contained in $\\Co_0$. In addition, in the\nset $\\{\\xi=(x,y)\\in Q_{\\sqrt{\\varrho}}\\setminus (P_M\\cup\nQ_{2\\sqrt{\\frac b3}\\varrho})\\;:\\; xy\\leq 0\\}$ the unstable direction\nis contained in $\\Co_1$.\n\\end{lem}\n\nTo conclude we need to study what happens in a neighborhood of the\norigin of order $\\varrho$. It is necessary to \ndistinguish two possibilities: one can enter below the stable\nmanifold, and hence be confined in the fat sector, or one can enter\nabove the stable manifold, thereby being bound to the thin sector.\nWe will start with the easy case: the second.\n \n\\subsection{The cone field--Thin sector}\n\nIf $x>0$, as soon as the trajectory, at some time $n$, enters\n$Q_{2\\sqrt{\\frac b3}\\varrho}$ we have that \n$(1,u)\\in\\Co_0$ implies $u\\geq 2b\\varrho\\geq \\gamma_u'(x_n)$.\nLet us consider in $Q_{2\\sqrt{\\frac b3}\\varrho}$ the cone field\n$\\Co_2:=\\{(1,u)\\;|\\; L\\bar u(\\xi)\\geq u(x)\\geq \\gamma_u'(x)\\}$. Note\nthat, upon entering in $Q_{2\\sqrt{\\frac b3}\\varrho}$ such a cone contains \n$\\Co_1$.\\footnote{Note that, in such a case, the trajectory cannot enter in $P_M$.}\nNow\n\\[\nF(x,u)\\geq F(x,\\gamma_u'(x))=\\gamma_u'(x+h(x)+\\gamma_u(x))\\geq\n\\gamma_u'(x+h(x)+y), \n\\]\nwhere we have use that $\\gamma_u''(x)\\geq 0$ for $x\\in[0,\\varrho]$,\nprovided $\\varrho$ has been chosen small enough.\n\\begin{lem}\n\\label{lem:conefield-two}\nIn the region $Q_{2\\sqrt{\\frac b3}\\varrho}\\setminus(\nP_M\\cup\\{\\xi=(x,y)\\in\\To^2\\;:\\; y\\geq \\gamma^u(x) \\text{ for }x>0;\ny\\leq \\gamma^u(x) \\text{ for }x<0\\})$ the\nunstable direction is contained in the cone field $\\Co_2$. \n\\end{lem}\nNote that the above lemma suffices for trajectories in the thin sector.\nThe situation it is not so simple in the fat sector since the lower\nbound would deteriorate to zero. A more detailed analysis is needed.\n\nFor each $\\xi=(x,y)\\in\\To^2$, for which the unstable direction is defined,\nlet $(1,u(\\xi))$ be the vector in the unstable direction. Define then\n$\\lambda_{u,n}(\\xi, u)$ and $F_n(\\xi,u)$ as in formulae \\eqref{eq:unsdef}\nand \\eqref{eq:unstreg} and similarly define the stable\nquantities. That is\n\\begin{equation}\\label{eq:constex}\n\\begin{split}\n&D_{T^{-n}\\xi} T^{n}(1,u)=:\\lambda_{u,n}(\\xi,u)(1,F_n(\\xi,u))\\\\\n&D_{\\xi} T^{-n}(1,-v)=:\\mu_{s,n}(\\xi,v)(1,-F^-_n(\\xi,v))\n\\end{split}\n\\end{equation}\n\n\\subsection{The cone field--Fat sector}\nFirst of all notice that the trajectory can enter\n$Q_{2\\sqrt {\\frac b3} \\varrho}$ either in $P_M$ or outside.\nSince the cone field $\\Co_2$ for $x\\geq\n0$, $\\xi\\not\\in P_M$ contains the unstable vector (Lemma\n\\ref{lem:conefield-two}), we have a good control on the \nunstable vector in both cases until we enter in $P_M$. \nUpon entering $P_M$, we will obtain a very sharp control on the\nevolution of the edges of the cone. Let $\\xi\\not\\in P_M$, $T\\xi\\in\nP_M$, and let $\\ell_+-1>0$ be the smallest integer such that\n$\\xi_n\\not\\in P_M$. By equation \\eqref{eq:unsdef}, we have\n\\[\nu_n:=F_n(\\xi_n,u)=\\sum_{i=1}^{n}\n\\lambda_{u,i}(\\xi_{n-i},u_{n-i})^{-1}h'(x_{n-i})+\\lambda_{u,n}(\\xi_n,u)^{-1}\nu. \n\\]\nThen, for each $n< \\ell_+$, holds true\n\\[\nu_n\\leq \\sum_{i=1}^n h'(x_{n-i})+u\\leq \\frac {\\operatorname{C} }{M} \n\\left|\\sum_{i=1}^n y_i\\right|+u. \n\\]\nBy Lemma \\ref{lem:zdyn2}-(3),(5), it follows that\nwe have, for $u\\in\\Co_2(\\xi)$,\n\\[\nu_n\\leq \\operatorname{C}_+\\sqrt{|y_n|}.\n\\]\nMoreover, remembering \\eqref{eq:unsdef} and that $u\\in\\Co_2(\\xi)$, yields\n\\[\nu_n\\geq e^{-2n\\operatorname{C}_+\\sqrt{|y_n|}}u\\geq \\operatorname{C} u\\geq \\operatorname{C}_-\\sqrt{|y_n|}. \n\\]\nConsequently, if for $\\xi=(x,y)$ we define the cone\n$\\Co_3(\\xi)=\\{\\operatorname{C}_-\\sqrt{|y|}\\leq u\\leq \\operatorname{C}_+\\sqrt{|y|}\\}$., then the\nabove results can be written as follows.\n\\begin{lem}\n\\label{lem:conefield-three}In $P_M$ the unstable direction\nis contained in the cone field $\\Co_3$.\n\\end{lem}\nFinally we have to follow the trajectory outside $P_M$ until it exits\nfrom $Q_{2\\sqrt{\\frac b3}\\varrho}$. The upper bound can be treated as\nbefore. Not so for the lower bound.\n\nLet $\\xi=(x,y)$ be a point in the\nfat sector, $x\\leq 0$, $x_{-1}\\geq 0$. Then, remembering subsection\n\\ref{sub:nearfix}, let $E:=H(x,y)$, $u_0=0$ and $u_{n+1}:=F(x_n,\nu_n)$. Clearly, $D_{(x,y)}T^n\\Co_+\\subset\\{(1,u)\\in\\R^2\\;|\\; u\\geq\nu_n\\}\\cup\\{0\\}$.\n\\begin{lem}\n\\label{lem:flowdir}\nIn the situation described above, for each $n\\in\\N$, holds true\n\\[\nF(x_n,\\Upsilon_E'(x_n))-\\Upsilon_E'(x_{n+1})=\\Or(|y_n|^{3\/2}).\n\\]\n\\end{lem}\n\\begin{proof}\nNotice that, since the trajectory lies below the unstable manifold, \n$|y|\\geq \\operatorname{C} x^2$. It is then convenient to keep track of the orders\nof magnitude only in terms of powers of $y$.\n\\[\nF(x_n,\\Upsilon_E'(x_n))=\\Upsilon_E'(x_n)-\\Upsilon_E'(x_n)^2\n+h'(x_n)+\\Or(|y_n|^{3\/2}).\n\\]\nOn the other hand, differentiating \\eqref{eq:upsi}, one gets \n\\[\n\\Upsilon_E'(x)=\\frac{h(x)}{\\Upsilon_E(x)}\n-\\frac{h(x)^2}{2\\Upsilon_E(x)^2} -\\frac{h'(x)}2+\\Or(|\\Upsilon_E|^{3\/2}). \n\\]\nAccordingly, by \\eqref{eq:dyny}, \n\\[\n\\begin{split}\n\\Upsilon_E'(x_{n+1})&=\\frac{h(x_n)+h'(x_n)\\Upsilon_E(x_n)}{\\Upsilon_E(x_n)+h(x_n)}\n-\\frac{h(x_n)^2}{2\\Upsilon_E(x_n)^2}-\\frac{h'(x_n)}2 \n+\\Or(|y_n|^{3\/2})\\\\\n&=\\Upsilon_E'(x_{n})-\\Upsilon_E'(x_{n})^2 +h'(x_n)+\\Or(|y_n|^{3\/2}),\n\\end{split}\n\\]\nfrom which the Lemma easily follows.\n\\end{proof}\nSince $F$ is a contraction in $u$, \nwe can estimate\n\\[\n\\begin{split}\n|u_n-\\Upsilon_E'(x_n)|&=|F(x_{n-1}, u_{n-1})-F(x_{n-1},\\Upsilon_E'(x_{n-1}))|\n+\\Or(|y_{n-1}|^{3\/2})\\\\\n&\\leq |u_{n-1}-\\Upsilon_E'(x_{n-1})|+\\Or(|y_{n-1}|^{3\/2})\\\\\n&\\leq|\\Upsilon_E'(x_0)|+\\Or\\left(\\sum_{k=0}^{n-1}|y_k|^{3\/2}\\right)\\\\\n&=\\Or\\left(|y_0|+\\sum_{k=0}^{n-1}\\sqrt{|y_k|}(x_{k}-x_{k+1})\\right)\n=\\Or(|y_n|).\n\\end{split}\n\\]\nWe have thus proved that there exists a constant $\\operatorname{C}_0>0$ such that\n\\begin{equation}\\label{eq:fatcone}\nu_n\\geq \\Upsilon_E'(x_n)-\\operatorname{C}_0\\Upsilon_E(x_n).\n\\end{equation}\nHence outside $P_M$ the image of the cone will belong to the\ncone field $\\Co_3:=\\{(1,u)\\in\\R^2\\;|\\; u(\\xi)\\geq\n\\Upsilon'_{E(\\xi)}(x)-\\operatorname{C}_0 \\Upsilon_{E(\\xi)}(x)\\}$. Note that, upon exiting $P_M$,\n$\\Upsilon_E'(x_n)-\\operatorname{C}_0\\Upsilon_E(x_n)\\geq \\operatorname{C}_-'\\sqrt{|y_n|}$,\nprovided $\\varrho$ is chosen small enough.\nThe Proposition follows by choosing $\\varrho$ small enough and\nremembering Lemmata \\ref{lem:conefield-one}, \\ref{lem:conefield-two}\nand \\ref{lem:conefield-three}.\n\n\n\\section{An a priori expansion bound}\n\\label{sec:expansion}\n\nThe results of the previous section allow to obtain the following nice\nestimate on the expansion in the system.\n\\begin{lem}\\label{lem:expansion}\nThere exists $K>0$ such that, for each\n$\\xi=(x,y)\\in\\To^2\\backslash \\{0\\}$, $n\\in\\N$ and\n$(1,u)\\in\\Co_*(T^{-n}\\xi)=\\Co_*(\\xi_{-n})$, holds true \n\\[\n\\lambda_{u,n}(\\xi,u)\\geq e^{-K|x|} \\left(K^{-1}\n|x|n+1\\right)^2 . \n\\]\n\\end{lem}\n\\begin{proof}\nLet us fix $\\delta>0$. On the one hand, if the trajectory lies outside\nof $Q_\\delta$, then \nwe have an exponential expansion, on the other hand, if the backward\ntrajectory enjoys $|x_{-n}|\\geq |x_0|$, then equation \\eqref{eq:unsdef}\nand Proposition \\ref{prop:dista} imply\n\\begin{equation}\n\\label{eq:trivialexp}\n\\lambda_{u,n}(\\xi,u)\\geq (1+K_-|x_0|)^n\\geq e^{-K|x_0|} \\left(K^{-1}\n|x_0|n+1\\right)^2 .\n\\end{equation}\nWe say that the backward orbit of $\\xi$ (up to time $n$) passes $p$ times \nthru\n$Q_\\delta$ if $\\{0\\leq k\\leq n\\;:\\;\\xi_{-k}\\in Q_\\delta\\}$ consists of\n$p$ intervals. The Lemma holds for orbits that pass zero-times thru\n$Q_\\delta$. Suppose it holds for orbits that pass $p$ times. Let\n$\\xi_{-n}\\in Q_\\delta$ and let $m0$ the\nbackward trajectory increases the $x$ coordinate. In such cases we\nhave\\footnote{Again, $E$ is chosen to be the energy associated\nto the point of the orbit closer to the origin.} \n\\[\n\\begin{split}\n\\lambda_{u,n}(\\xi,u)&=\\prod_{j=1}^n(1+h'(x_{-j})+u_{-j})\\geq\n\\prod_{j=1}^n(1+\\Upsilon_E'(x_{-j})-\\operatorname{C}_0\\Upsilon_E(x_{-j}))\\\\\n&\\geq e^{\\sum_{j=1}^n\\Upsilon_E'(x_{-j})-2\\operatorname{C}_0\\Upsilon_E(x_{-j})}\\\\\n&\\geq e^{-\\sum_{j=1}^n\\frac{\\Upsilon_E'(x_{-j})}{\\Upsilon_E(x_{-j})}\n(x_{-j-1}-x_{-j}) -3\\operatorname{C}_0\\sum_{j=1}^n(x_{-j-1}-x_{-j})} \\\\\n&\\geq e^{-\\operatorname{C}|x_0|}e^{-\\int_{x_0}^{x_{-n}}\\frac{\\Upsilon_E'(z)}{\\Upsilon_E(z)}dz}\n= e^{-\\operatorname{C}|x_0|}\\frac{\\Upsilon_E(x_0)}{\\Upsilon_E(x_{-n})}.\n\\end{split}\n\\]\n\nLet $n_*\\in\\N$ be the last integer for which $|G(x_{-n})|\\geq\nE$, then for $n\\leq n_*$ we have \n\\[\n\\lambda_{u,n}(\\xi)\\geq e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac\n{E+G(x_0)}{E+G(x_{-n})}} \n\\geq e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac\n{G(x_{-n})+G(x_0)}{2G(x_{-n})}}.\n\\]\nOn the other hand comparing the backward motion with the\nbackward motion on the stable manifold, as we did before with the\nunstable,\\footnote{Here we use the inequality \n\\[\n\\sqrt{\\frac{1+(1+a)^4}2}\\geq (1+\\frac a2)^2.\n\\]\n}\n\\[\n\\lambda_{u,n}(\\xi,u)\\geq\ne^{-\\operatorname{C}|x_0|}\\sqrt{\\frac{1+(n|x_0|\\operatorname{C}+1)^4}2}\n\\geq e^{-\\operatorname{C}|x_0|}(1+n|x_0|\\operatorname{C})^2.\n\\]\nNext, let us consider $n\\in\\{n_*,\\dots, m\\}$, where $m$ is the larger\ninteger such that $x_{-m}\\leq 0$, we have $2\\sqrt E\\geq \n\\Upsilon_E(x_{-n})\\geq \\sqrt {2E}$. \n\\[\n\\begin{split}\n\\lambda_{u,n-n_*}(\\xi,u)\n&\\geq e^{-\\operatorname{C}_2|x_{n_*}|}\ne^{-\\int_{x_{-n_*}}^{x_{-n}}\\frac{\\Upsilon_E'(z)}{\\Upsilon_E(z)}dz} \n\\geq e^{-\\operatorname{C}|x_0|}\\frac{\\Upsilon_E(x_{-n_*})}\n{\\Upsilon_E(x_{-n})}\\\\\n&\\geq e^{-\\operatorname{C}|x_0|}\\sqrt{\\frac 32}\\geq e^{-\\operatorname{C}|x_0|}(1+\\operatorname{C}\n|x_{-n_*}|(n-n_*))^2, \n\\end{split}\n\\]\nwhere, in the last line, we used Lemma \\ref{lem:zdyn2}-(5).\nBy symmetry it will be enough to wait another time $m$ to have\n$|x_{-2m}|\\geq \\frac 12|x_0|$, after which the expansion is assured by\nthe estimate \\eqref{eq:trivialexp}.\n\\end{proof}\n\nNext we need to have similar estimates for the stable contraction. By\n\\eqref{eq:constex} \n\\begin{equation}\\label{eq:stabdef}\n\\begin{split}\n\\mu_s(v)&:=1+v=\\mu_{s,1}(\\xi,v)\\\\\nF^-_1(\\xi,v)&=h'(x-y)+\\frac v{1+v}.\n\\end{split}\n\\end{equation}\nIt is immediate to check that\n$D_{(x,y)}T^{-1}(1,-v)=\\mu_s(v)(1,-F^-_1((x,y),v))$ and\n$\\mu_{s,n}(\\xi,v_{0}):=\\prod_{i=0}^{n-1}\\mu_s(v_{-i})$, where \n$v_0=v$ and $v_{-i-1}:=F^-_1(T^{-i}\\xi,v_{-i})$.\n\nAn interesting way to transform information on expansion into information on\ncontraction is to use area preserving. \n\\begin{lem}\\label{lem:areap}\nLet $\\xi\\in\\To^2$, then for each $n\\in\\N$, $u,v\\geq 0$ let $u_{-n}=u$,\n$v_0=v$, $\\xi_{-n}=T^{-n}\\xi$ and $u_{-k+1}=F(\\xi_{-k+1}, u_{-k})$,\n$v_{-k-1}=F^-(\\xi_{-k},v_{-k})$. Then\n\\[\n\\mu_{s,n}(\\xi, v_0)(v_{-n}+u_{-n})=\\lambda_{u,n}(\\xi, u_{-n})(u_0+v_0).\n\\]\n\\end{lem}\n\\begin{proof}\nCalling $\\omega$ the standard symplectic form we have\n\\[\n\\begin{split}\n\\mu_{s,n}(\\xi,v_0)(v_{-n}+u_{-n})&=\\omega(D_{\\xi}T^{-n}(1,-v_0),(1,u_{-n}))\\\\\n&= \\omega((1,-v_0), D_{\\xi_{-n}} T^n(1,u_{-n}))=\\lambda_{u,n}(\\xi,\nu_{-n})(u_0+v_0). \n\\end{split}\n\\]\n\\end{proof}\nThe following is an immediate corollary of Lemmata \\ref{lem:areap} and\n\\ref{lem:expansion}. \n\\begin{cor}\\label{lem:contraction}\nFor each $\\xi=(x,y)\\in \\To^2$ and $n\\in\\N$ holds\n\\[\n\\mu_{s,n}(\\xi, v_0)\\geq e^{-\\operatorname{C}|x_{0}|} (\\operatorname{C}^{-1}|x_{0}|n+1)^2\\frac\n{u_0+v_0}{u_{-n}+v_{-n}} \\quad\\forall n\\in\\N.\n\\]\n\\end{cor}\n\nAll the other expansion estimates can be obtained by reversibility.\n\n\\section{Distributions--regularity}\n\\label{sec:regularity}\n\nLet $(1,u(\\xi)),\\,(1,-v(\\xi))$ be the unstable and stable directions,\nrespectively. We will then use the short hand\n$\\lambda_{u,n}(\\xi):=\\lambda_{u,n}(\\xi,u(\\xi_{-n}))$ and\n$\\mu_{s,n}(\\xi):=\\mu_{s,n}(\\xi,v(\\xi_{n}))$. \n\\begin{lem}\\label{lem:distreg0}\nThe unstable distribution is continuous in $\\To^2$.\n\\end{lem} \n\\begin{proof}\nNotice that, for $\\xi=(x,y)$, $\\xi_n:=T^n\\xi$, iterating formula\n\\eqref{eq:unsdef}, in analogy with \\eqref{eq:unstreg}, holds true\n\\begin{equation}\\label{eq:unstreg-bis}\n\\begin{split}\nu(x)-u(z)=&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}(u(x_{-n})-u(z_{-n}))\\\\\n&\\ \\ +\\sum_{k=1}^{n}\\lambda_{u,k}(x)^{-1}\\lambda_{u,k}(z)^{-1}\n(h'(x_{-k})-h'(z_{-k}))\n\\end{split}\n\\end{equation}\nBy Lemma \\ref{lem:expansion}, we can take the limit $n\\to\\infty$ in\nthe above formula provided $x\\neq 0$, and obtain a\nuniformly convergent series from which the continuity follows.\nIf $\\xi\\neq 0$ then $x_{-1}\\neq 0$ and \\eqref{eq:unsdef} implies\n\\begin{equation}\n\\label{eq:uma}\nu(\\xi)=\\lambda_{u}(\\xi_{-1})^{-1}h'(x_{-1})\n+\\lambda_{u}(\\xi_{-1})^{-1}u(\\xi_{-1}) ,\n\\end{equation}\nhence the continuity at $\\xi\\neq 0$ follows. We are left with the\ncontinuity at the origin, but this is already implied by Proposition\n\\ref{prop:dista}. \n\\end{proof}\n\nThis means that we can extend the invariant unstable distribution\n(that, up to now, where defined--by Pesin theory--only almost everywhere) to a\ncontinuous everywhere defined vector field. The same statement holds\nfor the stable vectors by reversibility.\n\nGiven a continuous vector field there exists integral curves. Since\nwe do not know yet if the vector fields are Lipschitz, it does not\nfollows automatically that from a given point there exits only one\nintegral curve, yet this follows by standard dynamical\narguments. Clearly such integral curves are nothing else than the\nstable and unstable manifolds that are therefore everywhere\ndefined. In addition, remember that, by general hyperbolic theory, the\nfoliations are absolutely continuous, it follows that the above\neverywhere defined foliations are continuous. Unfortunately, for the\nfollowing much sharper regularity information is needed, this is\nobtained in the rest of the section.\n\nLet us call $\\partial^u, \\partial^s$ the derivative along the unstable\nand the stable vector fields, respectively. \n\n\\begin{lem}\\label{lem:uderbound}\nThe vector field $u$ is $\\Co^1$ along the unstable manifolds, apart from the\norigin, moreover\n\\[\n|\\partial^u\nu(\\xi)|\\leq C\\quad\n\\forall \\xi\\neq 0. \n\\]\n\\end{lem}\n\\begin{proof}\nIf $\\xi$ is outside of a \nneighborhood of the origin of size $\\delta$, then by Lemma\n\\ref{lem:expansion}, \\eqref{eq:unstreg-bis} we have, in analogy with\nthe arguments leading to \\eqref{eq:unstder},\n\\begin{equation}\n\\label{eq:uder-u-bond}\n|\\partial^u\nu(\\xi)|=\\left|\\sum_{k=1}^{\\infty}\\lambda_{u,k}(x)^{-3}h''(x_k)\\right| \\leq \\operatorname{C}\n\\sum_{n=0}^{\\infty}(\\delta n+1)^{-6}\\leq \\operatorname{C}. \n\\end{equation}\nSince the series converges uniformly the $\\Co^1$ property follows. To\nobtain a uniform bound more work is needed.\nIf $|\\xi|<\\delta$, formula \\eqref{eq:uma} implies\n\\[\n|\\partial^u u(\\xi)|\\leq\\lambda_u(\\xi_{-1})^{-3}|h''(x_{-1})|+\n\\lambda_u(\\xi_{-1})^{-3} |\\partial^u u(\\xi_{-1})|=:\\Psi(\\xi_{-1},\n|\\partial^u u(\\xi_{-1})|).\n\\]\nA simple computation, remembering Proposition \\ref{prop:dista}, shows\nthat \n\\[\n\\Psi(\\xi,\\varrho)\\leq7b|x|+\\frac{\\varrho}{1+3|u(\\xi)|}\\leq\n7b|x|+\\frac{\\varrho}{1+3K_-|x|} \\leq \\varrho ,\n\\]\nprovided $\\varrho\\geq \\frac{7b(1+3K_-)}{3K_-}$. Accordingly, for $\\rho$\nlarge enough, we have \n$|\\partial^u u(\\xi)|\\leq \\rho$, for all $\\xi$. \n\\end{proof}\n\nIt remains to investigate the regularity of the unstable distribution\nalong the stable direction.\n\n\\begin{lem}\\label{lem:distreg1}\nThe unstable distributions are $\\Co^1$ along stable manifolds, apart\nfrom the origin. Moreover\n\\[\n|\\partial^s u(\\xi)|\\leq \\operatorname{C} \\quad \\forall \\xi\\neq 0.\n\\]\n\\end{lem} \n\\begin{proof}\nLet us fix some arbitrary neighborhood of the origin.\nLet $x,z\\in W^s$ outside such a neighborhood. Let $W^s_0$ be the\npiece of stable manifold between such two points. Clearly\n$W^s_{n}:=T^nW^s_0$ grows for negative $n$. Let $n(x,z)$ be the largest\ninteger for which $|W^s_{-n}|\\leq |W^s_0|^{\\frac 14}$. Our first\nresult is a distortion bound.\n\\begin{sublem}\\label{slem:dist}\nFor each $n\\leq n(x,z)$ and $\\xi\\in\nW^s_{0}$, holds\n\\[\n\\operatorname{C}^{-1}\\frac{|W^s_{-n}|}{|W^s_{0}|}\\leq \\mu_{s,n}(\\xi)\\leq\n\\operatorname{C}\\frac{|W^s_{-n}|}{|W^s_{0}|} .\n\\]\n\\end{sublem}\n\\begin{proof}\nIf the backward orbit spends at least half of the time outside the\nneighborhood, then $W^s_{-n}$\ngrows exponentially fast, hence $n(x,z)\\leq \\operatorname{C}\\ln|W^s_0|^{-1}$ and\n$\\sum_{i=0}^{n(x,z)}|W^s_{-i}|\\leq \\operatorname{C}$. If this is not the case,\nthe worst possible situation is when $W^s_{-m}$ is the closest\nto the origin and all the trajectory lies in the neighborhood. In such a\ncase, letting $m:=n(x,z)$, \n\\[\n|W^s_{m}|=\\int_{W^s_0}\\mu_{s,m}(z)dz\\geq\n\\int_{W^s_0}\\frac{\\theta(z)(\\operatorname{C}^{-1}|z|m+1)^2}{2\\theta(T^{-m}z)} dz,\n\\]\nwhere $\\theta(\\zeta)=u(\\zeta)+v(\\zeta)$ is the separation between the\nstable and the unstable directions at the point $\\zeta$ and we have\nused Lemma \\ref{lem:contraction}. Now Proposition \\ref{prop:dista} and\nLemma \\ref{lem:zdyn2}-(1) imply $\\theta(T^{-m}z)\\leq \\operatorname{C} m^{-1}$\noutside the parabolic sector, while Lemma \\ref{lem:zdyn2}-(3,4,5) show\nthat the same estimates remain in $P_M$ as well. Accordingly,\n\\[\n|W^s_{-m}|\\geq \\operatorname{C} m^3|W^s_0|.\n\\]\nThat is $m\\leq \\operatorname{C} |W^s_0|^{-\\frac 14}$, and\n\\[\n\\sum_{i=0}^{n(x,z)}|W^s_{-i}|\\leq n(x,z)|W^s_0|^{\\frac 14}\\leq \\operatorname{C}.\n\\]\nThe above estimate readily implies that, for each $\\xi,\\eta\\in W^0_s$,\n\\[\ne^{-\\operatorname{C} |W^s_{-i}|}\n\\leq\\frac{\\mu_s(\\xi_{-i},v(\\xi_{-i}))}{\\mu_s(\\eta_{-i},v(\\eta_{-i}))}\n\\leq e^{\\operatorname{C} |W^s_{-i}|},\n\\]\nwhere we have used Lemma \\ref{lem:uderbound} for the stable manifold.\nAccordingly,\n\\[\ne^{-\\operatorname{C}\\sum_{i=0}^m |W^s_{-i}|}\n\\leq\\frac{\\mu_{s,n}(\\xi)}{\\mu_{s,n}(\\eta)}\n\\leq e^{\\operatorname{C} \\sum_{i=0}^m|W^s_{-i}|},\n\\]\nfrom which the Lemma readily follows.\n\\end{proof}\nBy Lemma \\ref{lem:areap} it follows, letting again $m:=n(x,z)$,\n\\[\n\\begin{split}\n\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\n&=\\lambda_{u,m}(x)^{-1}\\sqrt{\\lambda_{u,m}(z)^{-2}}\\\\\n&\\leq\n\\operatorname{C}\\left(\\theta(x_{-m})\\mu_{s,m}(x)\n\\sqrt{\\lambda_{u,m}(z)\\theta(z_{-m})\\mu_{s,m}(z)}\\right)^{-1}\n\\end{split}\n\\]\nAs before the worst case is clearly when $W^s_{-m}$ is the closest to\nthe origin.\nIn such a case, consider that at least one of the two end points of\n$W^s_{-n(x,z)}$ must be at a distance $\\operatorname{C} |W^s_{-n(x,z)}|$ from the\nfixed point, let us say $T^{-n(x,z)}z$, hence\n$\\theta(T^{-n(x,z)}z)\\geq \\operatorname{C}|W^s_{-n(x,z)}|$. In addition,\n$\\theta(x_{-m})\\geq \\operatorname{C} m^{-1}$. Indeed, this follows from Lemma\n\\ref{lem:zdyn2}-(3,4,5) if the trajectory ends in $P_M$. If the\ntrajectory lies outside $P_M$ then it approaches the origin slower than\nthe dynamics $x-\\operatorname{C} x^2$, which implies $x_{-m}\\geq \\operatorname{C}\nm^{-1}$. Furthermore by using the above facts, Lemma \\ref{lem:expansion}, Sub-lemma\n\\ref{slem:dist} and the definition of the stopping time $m$ yields\n\\[\n\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\\leq \\operatorname{C}|x-z|^{\\frac\n34}m\\frac 1{m} \\sqrt{\\frac{|x-z|^{\\frac \n34}}{|x-z|^{\\frac 14}}} \\leq \\operatorname{C}|x-z|.\n\\]\nSince we know that $u$ is a uniformly continuous function it follows\n\\[\n\\lim_{z\\to\nx}\\lambda_{u,m}(x)^{-1}\\lambda_{u,m}(z)^{-1}\n\\frac{|u(T^{-m}x)-u(T^{-m}z)|} \n{|x-z|}=0.\n\\]\nAccordingly, by formula \\eqref{eq:unstreg}, \n\\[\n\\begin{split}\nu'(x)=&\\lim_{z\\to x}\n\\sum_{n=0}^{m}\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\n\\frac{h'(T^{-n}x)-h'(T^{-n}z)}{x-z}\\\\\n=& \\lim_{z\\to x}\n\\sum_{n=0}^{m}\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\nh''(T^{-n}\\zeta_{n}) \\mu_{s,n+1}(\\zeta_n),\n\\end{split}\n\\]\nfor some $\\zeta_n\\in W_0$.\nBut $|h''(T^{-n}\\zeta_n)|\\leq \\operatorname{C}\\theta(T^{-n}z)$, hence\n\\[\n\\begin{split}\n&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\\theta(T^{-n}x)\n\\mu_{s,n+1}(x)\\leq \\operatorname{C}\\lambda_{u,n}(z)^{-1}\\\\\n&\\lambda_{u,n}(x)^{-1}\\lambda_{u,n}(z)^{-1}\\theta(T^{-n}z)\n\\mu_{s,n+1}(z)\\leq \\operatorname{C}\\lambda_{u,n}(x)^{-1}.\n\\end{split}\n\\]\nRemembering Sub-Lemma \\ref{slem:dist} the uniform convergence of the\nseries follows and yields the formula\n\\begin{equation}\n\\label{eq:uder-s-bound}\n\\partial^s u(x)=\\sum_{n=0}^{\\infty}\\lambda_{u,n}(x)^{-2}\\mu_{s,n+1}(x)\nh''(T^{-n}x).\n\\end{equation}\nGiven the arbitrariness of the neighborhood of zero, the above formula\nholds for each $x\\neq 0$ and, since the series converges uniformly,\nthe $\\Co^1$ property follows. We can now conclude the Lemma.\nBy Lemma \\ref{lem:areap} follows\n\\[\n\\begin{split}\n|\\partial^s u(x)|&\\leq\\operatorname{C} \\sum_{n=0}^\\infty\n\\lambda_{u,n}(x)^{-2}\\mu_{s,n}(x)\\theta(T^{-n}x) \\\\\n&\\leq \\operatorname{C} \\sum_{n=0}^\\infty\\lambda_{u,n}(x)^{-1}\\theta (x)\n\\leq \\operatorname{C} \\sum_{n=0}^\\infty\\frac{|x|}{(|x|n+1)^2}\\leq \\operatorname{C} .\n\\end{split} \n\\]\n\\end{proof}\n\\begin{rem}\nNotice that the symmetrical statements follow by reversibility.\n\\end{rem}\nThe final result on the regularity of the foliations can be stated as\nfollows.\n\\begin{lem}\n\\label{lem:c1-reg}\nThe stable and unstable vector fields are $\\Co^1(\\To^2\\setminus\\{0\\})$ and, more\nprecisely, for each $\\xi\\in\\To^2\\setminus\\{0\\}$,\n\\[\n|Du(\\xi)|\\leq \\operatorname{C} \\theta(\\xi)^{-1}.\n\\]\n\\end{lem}\n\\begin{proof}\nThe $\\Co^1$ property follows from Lemma 19.1.10 of \\cite{KH}.\nThen the size of the derivative can be easily estimated by the size of\nthe partial derivatives in the stable and unstable directions divided by\nthe angle between them.\n\\end{proof}\n\\begin{rem}\nIn fact, it is likely that with a little more work one can show that\nthe foliations are $\\Co^{\\frac 32-\\ve}$, but we do not investigate this\npossibility since it is not needed in the following.\n\\end{rem}\n\\section{Holonomy}\n\\label{sec:holo}\nThere exists $\\operatorname{C}_1>0$ such that, given two close by stable manifolds\n$W^s_1$, $W^s_2$ we can define the {\\em unstable holonomy}\n$\\Psi^u:W^s_1\\to W^s_2$ by $\\{\\Psi^u(\\xi)\\}:=W^u(\\xi)\\cap W^s_2$. \nLet $D_r:=\\{\\zeta=(z_1,z_2)\\in\\R^2\\;:\\;|z_1|\\leq r;|z_2|\\leq r^2\\}$.\n\\begin{lem}\\label{lem:holo}\nFor each $W^s_1,W^s_2$ disjoint from $D_r$ and $\\xi\\in W^s_1$, holds\n\\[\n|J\\Psi^u(\\xi)-1|\\leq \\operatorname{C} r^{-1}\\|\\Psi^u(\\xi)-\\xi\\|.\n\\]\nProvided $\\|\\Psi^u(\\xi)-\\xi\\|\\leq \\operatorname{C}_1 r$.\n\\end{lem}\n\\begin{proof}\nLet $\\gamma_s,\\tilde\\gamma_s:[-\\delta,\\delta]\\to\\R^2$ be $W^s_1,\nW^s_2$, respectively, parametrized by arc-length. Also, let\n$\\Gamma:[-\\delta,\\delta]^2\\to\\R^2$, be such that $\\Gamma(0,0)=\\xi$,\n$\\Gamma(s,0)=\\gamma_s(s)$ \nand $\\Gamma(s,t)$ be the unstable manifold, parametrized by\narc-length, of $\\Gamma(s,0)$ and, finally,\n$\\Gamma(0,\\rho):=\\Psi^u(\\xi)$. Note that $\\Gamma(s,t)$ can be obtained\nintegrating the unstable vector field starting from $\\Gamma(s,0)$,\nhence Lemma \\ref{lem:c1-reg} and the standard results on the continuity\nwith respect to the initial data imply $\\Gamma\\in\\Co^1$. By the\ntransversality of the stable and unstable manifolds there exist\n$\\tau,\\sigma:[-\\delta,\\delta]\\to\\R$ such that\n$\\Gamma(s,\\tau(s))=\\Psi^u(\\gamma_s(s))=\\tilde\\gamma(\\sigma(s))\\in\nW^s_2$. Calling $\\eta(s)$ the unit vector perpendicular to $\\tilde\n\\gamma'(s)$, by the implicit function theorem, it follows \n\\begin{equation}\n\\label{eq:implicit}\n\\begin{split}\n\\tau'(s)&=-\\frac{\\langle \\eta(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle}\n{\\langle\\eta(s), \\partial_t\\Gamma(s,\\tau(s))\\rangle}\\\\\n\\sigma'(s)&=\\langle\\tilde\\gamma_s'(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle\n-\\frac{\\langle\\tilde \n\\gamma'_s(s),\\partial_t\\Gamma(s,\\tau(s))\\rangle\\;\\langle\n\\eta(s),\\partial^s\\Gamma(s,\\tau(s))\\rangle} \n{\\langle\\eta(s), \\partial_t\\Gamma(s,\\tau(s))\\rangle}\n\\end{split}\n\\end{equation}\nwhere, clearly, $\\sigma'(s)=J\\Psi^u(\\gamma_s(s))$. Calling\n$v^u(\\eta)$, $\\eta\\in\\To^2$,\nthe unit vector in the unstable direction at $\\eta$ and $v^s(\\eta)$ the\nstable one, one has\n$\\partial_t\\Gamma(s,\\tau(s))=v^u(\\Gamma(s,\\tau(s)))$. On the other\nhand, setting $V(s,t):=\\partial^s\\Gamma(s,t)-v^s(\\Gamma(s,t))$, holds\n$V(s,t)=0$ for $t=0$, but for $t\\neq 0$, in general, it will be\n$V(s,t)\\neq 0$. Yet, it is possible to estimate it by differentiating\n$\n\\Gamma(s,t)=\\Gamma(s,0)+\\int_0^tv^u(\\Gamma(s,t'))dt'\n$\nwhich yields\n\\[\n\\partial^s\\Gamma(s,t)=v^s(\\Gamma(s,0))+\\int_0^t Dv^u(\\Gamma(s,t'))\n\\partial^s\\Gamma(s,t')dt'.\n\\]\nLemmata \\ref{lem:c1-reg} and \\ref{lem:distreg1}\nimply that $\\|Dv^u\\|\\leq \\operatorname{C} r^{-1}$ and $\\|Dv^u v^s\\|=|\\partial^s\nv^u|\\leq \\operatorname{C}$, hence \n\\[\n\\|V(s,t)\\|\\leq \\operatorname{C} r^{-1}\\int_0^t\\|V(s,t')\\|dt'+\\operatorname{C} t.\n\\]\nBy Gronwal, it follows, provided $t\\leq \\operatorname{C} \\rho$ and $\\rho\\leq \\operatorname{C}_1\nr$, for $\\operatorname{C}_1$ small enough,\n\\begin{equation}\n\\label{eq:Vest}\n\\|V(s,t)\\|\\leq \\operatorname{C} t.\n\\end{equation}\nAccordingly, by the second of \\eqref{eq:implicit} and \\eqref{eq:Vest},\nit follows\n\\[\n|\\sigma'(0)-1|\\leq \\operatorname{C} r^{-1}\\rho.\n\\]\n\\vskip-.5cm\n\\end{proof}\n\n\\section{Random perturbations}\n\\label{sec:random}\n\nThe density of a measure with respect to Lebesgue evolves as\n\\[\n\\Lp f:=f\\circ T^{-1}.\n\\]\nWe will then construct a random perturbation by introducing the\nconvolution operator\n\\begin{equation}\\label{eq:conv}\nQ_\\ve f(x):=\\int_{\\To^2}q_\\ve(x-y)f(y)dy.\n\\end{equation}\nWhere we assume\n\\begin{enumerate}\n\\item $q_\\ve(\\xi):=\\ve^{-2}\\bar q(\\ve^{-1}\\xi)$; $\\bar q\\in\\Co^{\\infty}(\\R^2,\\R^+)$;\n\\item $\\int_{\\R^2}\\bar q(\\xi)d\\xi=1$;\n\\item $\\bar q(\\xi)=0$ for each $\\|\\xi\\|\\geq 1$;\n\\item $\\bar q(\\xi)=1$ for each $\\|\\xi\\|\\leq \\frac 12$.\n\\end{enumerate}\nWe define then\n\\begin{equation}\\label{eq:random}\n\\Lp_\\ve:=Q_\\ve\\Lp^{n_\\ve},\n\\end{equation}\nwhere $n_\\ve$ will be chosen later.\n\nNotice that \n\\begin{equation}\\label{eq:ker1}\n\\Lp_\\ve^2f(x)=\\int_{\\To^4} q_\\ve(x-y)q_\\ve(T^{-n_\\ve}y-T^{n_\\ve}z)f(z)\ndz dy :=\\int_{\\To^2}\\Ka_\\ve(x,z)f(z)dz.\n\\end{equation}\n\nWe have thus a kernel operator that can be investigated with rather\ncoarse techniques. It turns out to be convenient to define the\nassociated kernel\n\\begin{equation}\\label{eq:kernel2}\n\\bar\\Ka_\\ve(x,z):=\\Ka_\\ve(x,T^{-n_\\ve}z)=\\int_{\\To^2}q_{\\ve}(x-y)\nq_\\ve(T^{-n_\\ve}y-z)m(dy). \n\\end{equation}\nFor further use let us define\n\\begin{equation}\\label{eq:levelset}\n\\begin{split}\n&D_r:=\\{z=(z_1,z_2)\\in\\To^2\\;|\\;|z_1|\\leq r;\\; |z_2|\\leq r^2\\}\\\\\n&B_r(\\xi):=\\{\\eta\\in\\To^2\\;;\\;\\|\\xi-\\eta\\|0$ such that, for each $\\delta0$ such that if $n_\\ve=\\operatorname{C}_3\\ve^{-\\frac 12}$ holds\n\\[\n\\bar\\Ka_\\ve(x,z)\\geq \\sigma\\quad \\forall x,z\\in\\To^2.\n\\]\n\\end{lem}\n\\begin{proof}\nIt is trivial to see that\n\\[\n\\bar\\Ka_\\ve(x,z)\\geq \\ve^{-4}m(B_{\\ve\/2}(x)\\cap T^{n_\\ve}B_{\\ve\/2}(z)).\n\\]\nAccordingly, by Lemma \\ref{lem:levelset-escape} there exists two\nballs, of radius $\\frac \\ve 8$, $\\bar B_1\\subset B_{\\ve\/2}(z)$ and\n$\\bar B_2\\subset B_{\\ve\/2}(x)$ that are outside of $D_{r}$, $r\\geq\n\\sqrt{\\frac \\ve 2}$, and whose images will be outside\nof a neighborhood of the origin or order one in a time less than $\\operatorname{C}\n\\ve^{-\\frac 12}$, forward and backward in time, respectively. Given two \nunstable manifolds in $B_1$ at a distance larger than $\\bar c r\\ve$, for\nsome appropriate $\\bar c$, then no stable \nmanifold will intersect both manifolds inside the ball $B_1$. We\ncan thus consider $\\operatorname{C} r^{-1}$ unstable manifolds such that no stable\nmanifolds intersect two of them in $B_1$. Around each such\nmanifold we can construct a strip by moving along the stable manifold\nby $\\operatorname{C} \\ve$. We obtain in this way $\\operatorname{C} r^{-1}$ disjoint strips\neach of area $\\operatorname{C} r\\ve^2$, whose union covers a fixed fraction of the area of\n$B_1$. After a time less that $\\operatorname{C} \\ve^{-\\frac 12}$ such strips will be outside a\nneighborhood of zero, their length may have increase considerable, if so\nwe will subdivide them into strips of length $\\ve$. Since now the\nstable and unstable manifold are at a fixed angle and by the usual\ndistortion arguments, such strips are essentially rectangular. At this point,\nby Lemma \\ref{lem:expansion}, it will suffice to wait a time\n$\\ve^{-\\frac 12}$ to insure that each such strip will acquire length\nat least $\\frac 12$ in the unstable direction. We thus iterate for such\na time and, if one strip becomes longer than one, we subdivide it into\npieces of length between $\\frac 12$ and one. Finally, fix some box\n$\\Lambda$ of some fixed size $C$ away from the origin with sides\napproximately parallel either to the stable or to the unstable\ndirections. \nBy mixing it suffices to wait a fixed time to be sure that a fixed\npercentage of each one of the above mentioned strips will intersect\nthe box. In addition, it is possible to insure that such strips cut\nthe box from one stable side to the other.\n\nWe can then write $m(B_{\\ve\/2}(x)\\cap\nT^{n_\\ve}B_{\\ve\/2}(z))=m(T^{-n_\\ve\/2}B_{\\ve\/2}(x)\\cap\nT^{n_\\ve\/2}B_{\\ve\/2}(z))$ since the same considerations done above for\nthe unstable manifold can be done, iterating \nbackward, for the stable manifold it follows that a fixed percentage of\n$T^{-n_\\ve\/2}B_{\\ve\/2}(x)$ and a fixed percentage of\n$T^{n_\\ve\/2}B_{\\ve\/2}(z))$ will intersect $\\Lambda$ and hence each\nother. In fact each one of the above constructed strips in the unstable\ndirection will intersects each one of the strips in the stable\ndirection. By the usual distortion estimates, this implies that the\nintersection among any two such strip has a measure proportional to the\nproduct of the measure of the two strips, hence\n\\[\nm(B_{\\ve\/2}(x)\\cap T^{n_\\ve}B_{\\ve\/2}(z))\\geq \\operatorname{C}\nm(B_{\\ve\/2}(x))m(B_{\\ve\/2}(z)),\n\\]\nand the lemma.\n\\end{proof}\n\n\\begin{lem}\\label{lem:l1}\nFor each $f\\in L^1$, $\\int f=0$ holds\n\\[\n\\|\\Lp_\\ve^n f\\|_1\\leq (1-\\sigma)^{n\/2}\\|f\\|_1.\n\\]\n\\end{lem}\n\\begin{proof}\nNote that $\\Lp_\\ve 1=\n\\Lp_\\ve^*1=1$ and let $\\M^+_\\ve=\\{x\\in\\To^2\\;|\\;\\Lp_\\ve^2 f\\geq 0\\}$;\n$\\M_+=\\{x\\in\\To^2\\;|\\; f\\geq 0\\}$, \nthen, since $\\int \\Lp_\\ve f=\\int f=0$,\n\\[\n\\begin{split}\n\\|\\Lp_\\ve^2 f\\|_1&=2\\int_{\\M^+_\\ve}dx\\,\\Lp_\\ve^2 f\n=2\\int_{\\M^+_\\ve}dx\\int_{\\To^2}dy\\, \\Ka_\\ve(x,\\,y) \nf(y)\\\\\n&=2\\int_{\\To^2}dy\\, f(y)\\int_{\\M^+_\\ve}dx[ \\Ka_\\ve(x,\\,y)-\\sigma]\\\\\n&\\leq 2\\int_{\\M_+}dy\\, f(y)\\int_{\\To^2}dx [\\Ka_\\ve(x,\\,y)-\\sigma]\n=2(1-\\sigma)\\int_{\\M_+}dy\\, f(y)\\\\\n&=(1-\\sigma)\\|f\\|_1.\n\\end{split}\n\\]\n\\end{proof}\nLet $v^{u,s}=(v^{u,s}_1,v^{u,s}_2)$ be the unit tangent vector fields in the\nunstable and stable direction, respectively. Clearly\n$|\\partial^u(\\Lp^i f)|\\leq |\\partial^u f|$, while \n$|\\partial^s(g\\circ T^i)\\leq |\\partial^s g|$.\n\\begin{lem}\\label{lem:appr}\nFor each $f,g\\in\\Co^{1}(\\To^2,\\R)$ holds\n\\[\n\\left|\\int Q_\\ve f g-\\int f g\\right|\\leq \\operatorname{C}\n\\ve\\{\\|f\\|_\\infty+\\|\\partial^u f\\|_{L^1(\\nu)}\\}\n\\{\\|g\\|_\\infty+\\|\\partial^s g\\|_{L^1(\\nu)}\\}, \n\\] \nwhere $\\nu$ is the measure defined by $\\nu(h):=\\int\nd\\rho \\int_{\\partial D_{\\rho}}h$.\n\\end{lem}\n\\begin{proof}\nIt is convenient to introduce the following change of\nvariables. For each $x,y\\in\\To^2$ close enough, let us call\n$[x,y]=W^u_\\delta(x)\\cap W^s_\\delta(y)$, note that by Lemma\n\\ref{prop:dista} such a point is\nalways well defined provided $d(x,y)\\leq \\operatorname{C} \\delta^2$. We consider then\nthe change of variable $\\Phi:\\To^2\\times \\To^2\\to\\R^2\\times\\To^2$,\n\\begin{equation}\\label{eq:changv}\n\\begin{split}\n\\xi:&=x-y\\\\\n\\eta:&=[x,y].\n\\end{split}\n\\end{equation}\nDue to the absolute continuity of the holonomies the above change of\nvariable is absolutely continuous. Clearly, $\\partial^u_x \\eta=0$ since\nmoving along $W^u(x)$ \ndoes not change the intersection point with $W^s(y)$. On the other\nhand $\\partial^s_x \\eta=J\\Psi^u_x v^s(\\eta)$, since moving $x$ along\n$W^s(x)$ moves $\\eta$ on $W^s(y)$ by an amount determined exactly by the unstable\nholonomy $\\Psi^u$ between $W^s(x)$ and $W^s(y)$. By similar arguments\nand a straightforward computations \n\\[\n\\begin{split}\n\\partial_{x_1}\\eta&=v_2^u(x)\\det\\begin{pmatrix} v^s(x)\n&v^u(x)\\end{pmatrix}^{-1}J\\Psi^u_x v^s(\\eta)\\\\\n\\partial_{x_2}\\eta&=v_1^u(x)\\det\\begin{pmatrix} v^u(x)\n&v^s(x)\\end{pmatrix}^{-1}J\\Psi^u_x v^s(\\eta)\\\\\n\\partial_{y_1}\\eta&=v_2^s(y)\\det\\begin{pmatrix} v^u(y)\n&v^s(y)\\end{pmatrix}^{-1}J\\Psi^s_y v^u(\\eta)\\\\\n\\partial_{y_2}\\eta&=v_1^s(y)\\det\\begin{pmatrix} v^s(y)\n&v^u(y)\\end{pmatrix}^{-1}J\\Psi^s_y v^u(\\eta).\n\\end{split}\n\\]\nIn fact, calling $\\theta(x)$ the sine of the angle between stable and\nunstable directions at the point $x$ and $v_\\perp$ the orthogonal\nunit vector to $v$, holds \n\\[\n\\begin{split}\n\\partial_x\\eta&=J\\Psi^u_x\\theta(x)^{-1}\\, |v^s(y)\\rangle\\langle v^u_\\perp(x)|\\\\\n\\partial_y\\eta&=J\\Psi^s_y\\theta(y)^{-1}\\, |v^u(x)\\rangle\\langle v^s_\\perp(y)|.\n\\end{split}\n\\]\nAccordingly,\n\\begin{equation}\\label{eq:jactot}\n\\begin{split}\nJ\\Phi&:=\\det\\begin{pmatrix}\n \\Id &-\\Id\\\\\n \\partial_{x}\\eta&\n \\partial_{y}\\eta \n \\end{pmatrix}=\\det\\begin{pmatrix}\n \\partial_{x}\\eta+\n \\partial_{y}\\eta \n \\end{pmatrix}\\\\\n&=J\\Psi^u_x\\, J\\Psi^s_y\\,\\theta(x)^{-1}\\theta(y)^{-1}\\langle v^s_\\perp(y),\\,\nv^u(x)\\rangle^2\\\\\n&=J\\Psi^u_x\\, J\\Psi^s_y\\,\\theta(x)^{-1}\\theta(y)^{-1}\\det\\begin{pmatrix}\n v^u(x)&v^s(y)\n \\end{pmatrix}^2. \n\\end{split}\n\\end{equation}\nBefore starting computing we need to collect some facts.\n\\begin{sublem}\n\\label{slem:domains}\nIf $x\\in\\partial D_{2\\rho}$, $\\rho\\geq r\\geq \\operatorname{C}_4\\sqrt\\ve$, then, for\n$C_4$ large enough,\n\\begin{enumerate}[i)]\n\\item if $\\|x-y\\|\\leq \\ve$, then $y\\not\\in D_\\rho$.\n\\item $\\|x-\\eta\\|\\leq \\frac\\ve\\rho$.\n\\item if $\\zeta\\in W^u(x)$ and $\\|\\zeta-x\\|\\leq\\frac\\ve\\rho$, then\n$\\zeta\\not\\in D_\\rho$.\n\\end{enumerate}\n\\end{sublem}\n\\begin{proof}\nThe first inequality follows since $D_{2\\rho}$ has a vertical size\n$4\\rho^2$. Thus $2\\rho^2-\\ve\\geq 2\\rho^2-\\frac\n1{\\operatorname{C}_4^2}\\rho^2\\geq\\rho^2$. Such an estimate and Proposition\n\\ref{prop:dista} imply that the angle between $W^u(x)$ and $W^u(y)$ is\nat least $4K_-\\rho^{-1}$, thus (ii). Finally, Proposition\n\\ref{prop:dista} implies that, if $\\zeta=(z_1,z_2)$,\n$z_2>2\\rho^2-\\frac{\\ve^2}{\\rho^2}\\geq\\rho^2$. \n\\end{proof}\nWe can now start computing the integral.\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy f(x)g(y)q_\\ve(x-y)&=\\int_{D_{\\operatorname{C}_4 r}^c}dx\\int_{\\To^2}dy\nf(x)g(y)q_\\ve(x-y) +\\Or(\\|f\\|_\\infty\\|g\\|_\\infty r^3)\\\\\n&\\geq\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)}\\!\\!\\!\\!\\!\\! d\\eta d\\xi\nf(x)g(y)q_\\ve(\\xi)J\\Phi^{-1}+ \n\\|f\\|_\\infty\\|g\\|_\\infty \\Or(r^3)\n\\end{split}\n\\]\nNext, from formula \\eqref{eq:jactot} and Sub-lemma \\ref{slem:domains}-$(i)$\nfollows \n\\begin{equation}\\label{eq:jac-bound}\n|J\\Phi(x,y)-1|\\leq \\operatorname{C}\\frac\\ve{\\rho^2}.\n\\end{equation}\nHence,\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy f(x)g(y)q_\\ve(x-y)\n&=\\int_{\\To^2} d\\eta\nf(\\eta)g(\\eta)+\\|f\\|_\\infty\\|g\\|_\\infty\\Or\\left(\\int_{D_r^c}|1-J\\Phi^{-1}|\\right)\\\\ \n&\\quad+\\Or\\left(\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} d\\eta d\\xi\n[f(x)-f(\\eta)]g(y)q_\\ve(\\xi)\\right) \\\\\n&\\quad+\\Or\\left(\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} d\\eta d\\xi\nf(\\eta)[g(y)-g(\\eta)]q_\\ve(\\xi)\\right)\\\\\n&\\quad +\\|f\\|_\\infty\\|g\\|_\\infty \\Or(r^3).\n\\end{split}\n\\]\nTo conclude we must compute the various error terms. \nFor each $f\\in L^\\infty$, holds\n\\[\n\\int_{D_R}dx\\,f(x)=\\frac 23\\int_0^Rd\\rho\\, \\frac \\rho{1+\\rho}\\int_{\\partial\nD_\\rho}ds\\, f(s).\n\\]\nRemembering\n\\eqref{eq:jac-bound} and applying Fubini\n\\[\n\\int_{D_r^c}|1-J\\Phi^{-1}|\\leq \\operatorname{C}\\|f\\|_\\infty\\int_r^1 d\\rho\\,\n\\rho^2\\frac\\ve{\\rho^2}\\leq \\operatorname{C}\\|f\\|_\\infty\\ve.\n\\]\nNext, let $\\gamma_u^\\eta:[-\\delta,\\delta]\\to\\To^2$ be the unstable manifold\nof $\\eta$, parametrized by arc-length, and let $s(\\eta,\\xi)$ be such\nthat $\\gamma_u^\\eta(s(\\eta,\\xi))=x$. Recalling Sub-lemma \\ref{slem:domains},\n\\[\n\\begin{split}\n\\left|\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! d\\eta d\\xi\\,\n[f(x)-f(\\eta)]g(y)q_\\ve(\\xi)\\right|&\\leq\n\\|g\\|_\\infty\\int_{\\Phi(D_{\\operatorname{C}_4 r}^c\\times \\To^2)} \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!d\\eta d\\xi\n\\int_0^{s(\\eta,\\xi)}dt\\,|\\partial^u f(\\gamma_u^\\eta(t))|q_\\ve(\\xi)\\\\\n&\\leq\\operatorname{C} \\|g\\|_\\infty\\int_{D_{r}^c\\times\n\\To^2} d\\eta' d\\xi\\, |\\partial^u f(\\eta')|\\frac{\\|\\xi\\|}{|\\theta(\\eta')|}q_\\ve(\\xi)\\\\\n&\\leq \\operatorname{C} \\|g\\|_\\infty\\int_0^1 d\\rho \\;\\ve\\int_{\\partial\nD_\\rho}|\\partial^uf|\\\\\n&\\leq \\operatorname{C} \\|g\\|_\\infty\\|\\partial^uf\\|_{L^1(\\nu)}\\ve.\n\\end{split}\n\\]\nAnalogously,\n\\[\n\\left|\\int_{D_{r}^c\\times \\To^2} d\\eta d\\xi\nf(\\eta)[g(y)-g(\\eta)]q_\\ve(\\xi)\\right|\\leq \\operatorname{C}\n\\|f\\|_\\infty\\|\\partial^sg\\|_{L^1(\\nu)}\\ve.\n\\]\nWe can finally collect all the above estimates and obtain\n\\[\n\\begin{split}\n\\int_{\\To^4}dxdy &f(x)g(y)q_\\ve(x-y)=\\int_{\\To^2} d\\eta\nf(\\eta)g(\\eta)\\\\\n&+(\\|f\\|_\\infty+\\|\\partial^uf\\|_{L^1(\\nu)})(\\|g\\|_\\infty\n+\\|\\partial^sg\\|_{L^1(\\nu)}) \\Or(r^3+\\ve)\n\\end{split}\n\\]\nfrom which the lemma follows by choosing $r=\\ve^{\\frac 13}$.\n\\end{proof}\n\n\n\\section{Decay of correlations}\n\\label{sec:decay}\nHere we put together the results of the previous section to prove\nTheorem \\ref{thm:main}. \n\nLet $f,g\\in\\Co^{1}(\\To^2,\\R)$, $\\int f=0$, then\n\\[\n\\begin{split}\n\\int\\Lp^{kn_\\ve}f g&=\\sum_{i=0}^{k-1}\\int\\Lp^{n_\\ve\ni}(\\Lp^{n_\\ve}-\\Lp_\\ve)\\Lp_\\ve^{k-i-1}f g+\\int\\Lp^k_\\ve fg\\\\\n&=\\sum_{i=0}^{k-1}\\int(\\Lp^{n_\\ve}-\\Lp_\\ve)\\Lp_\\ve^{k-i-1}f g\\circ T^{n_\\ve\ni}+\\Or(e^{-\\operatorname{C} k}\\|f\\|_1\\,\\|g\\|_\\infty)\\\\\n&=\\sum_{i=0}^{k-1}\\int(\\Id-Q_\\ve)\\Lp^{n_\\ve}\\Lp_\\ve^{k-i-1}f g\\circ T^{n_\\ve\ni}+\\Or(e^{-\\operatorname{C} k}\\|f\\|_1\\,\\|g\\|_\\infty) ,\n\\end{split}\n\\]\nwhere we have used Lemma \\ref{lem:l1}. To conclude, by using Lemma\n\\ref{lem:appr}, we need to estimate the $L^1$ norm of\n$\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)$. Since $|D\\Lp_\\ve^{j}f|\\leq \\operatorname{C}\n\\ve^{-1}|f|_\\infty$, for $j>0$,\n\\[\n\\begin{split}\n\\|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)\\|_{L^1(\\nu)}&\\leq\n\\int_0^{\\operatorname{C}_4\\sqrt \\ve}d\\rho\\int_{\\partial\nD_\\rho}|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)|\n+\\int_{\\operatorname{C}_4\\sqrt \\ve}^1d\\rho\\int_{\\partial\nD_\\rho}|\\partial^u(\\Lp^{n_\\ve}\\Lp_\\ve^{j}f)|\\\\\n&\\leq \\operatorname{C}\\|f\\|_\\infty \\int_0^{\\operatorname{C}_4\\sqrt \\ve}d\\rho\\,\\ve^{-1}\\rho+\n\\operatorname{C}|f|_\\infty\\int_{\\operatorname{C}_4\\sqrt \\ve}^1d\\rho\\int_{\\partial\nD_\\rho}\\frac{\\ve}{\\rho^2}\\ve^{-1}\\\\\n&\\leq \\operatorname{C} \\|f\\|_\\infty\\ln\\ve^{-1},\n\\end{split}\n\\]\nwhere we have used Lemma \\ref{lem:expansion} and Sub-lemma\n\\ref{slem:domains}.\\footnote{The above estimate is not sharp. With\nsome extra work one could avoid the $\\ln\\ve^{-1}$, yet this would not\nchange in any substantial way the result, so we chose to keep the\npresentation as short as possible.}\n\n\nThus\n\\begin{equation}\n\\label{eq:sharp}\n\\begin{split}\n\\left|\\int\\Lp^{n}f\ng\\right|=(\\|f\\|_{\\infty}+\\|\\partial^u f\\|_{L^1(\\nu)})\\|g\\|_{\\Co^1}\\Or(n\\ve^{\\frac\n32}\\ln\\ve^{-1}+ \ne^{-\\operatorname{C} n\\sqrt \\ve}).\n\\end{split}\n\\end{equation}\nClearly the best choice is $\\ve=\\operatorname{C} (n^{-1}\\ln n)^2$ which implies the\nTheorem.\n\n\n\n\\section{Lower bound}\n\\label{sec:lower}\nIn this section we prove a lower bound.\n\\begin{lem}\\label{lem:lower}\nIf there exists a sequence $\\gamma_n$ such that, for each\n$f,g\\in\\Co^1$ holds\n\\[\n\\left|\\int_{\\To^2}f\\circ T^{-n} g-\\int_{\\To^2}f\\int_{\\To^2}g\\right| \\leq\n(\\|\\partial^u f\\|_{L^1(\\nu)}+\\|f\\|_{L^1(\\nu)})|g|_{\\Co^1}\\gamma_n,\n\\]\nthen there exists $\\operatorname{C}>0$ such that\n\\[\n\\gamma_n\\geq n^{-2}\\operatorname{C}.\n\\]\n\\end{lem}\n\\begin{proof}\nLet $g\\geq 0$ be a smooth function supported away from zero (let us say that\nthe support of $g$ does not intersect $D_{\\frac 12}$). Next, let\n$\\xi_0=(x_0,y_0)=(\\frac 12,y_0)\\in W^u(0)$, $\\xi_n=T^{-n}\\xi_0$. For\neach point $\\eta$ in a neighborhood of $\\xi_n$ let $z_1$ be the\ndistance, along $W^u(0)$, between $\\xi_n$ and $W^u(0)\\cap W^s(\\eta)$,\nand $z_2$ the distance, along $W^s(\\xi_n)$, between $\\xi_n$ and\n$W^u(\\xi_n)\\cap W^s(\\eta)$. By construction \n$\\Xi_n(\\eta):=(z_1,z_2)$ is a map from a neighborhood of $\\xi_n$ to a\nneighborhood of the origin with the property that the map transforms\nthe stable and unstable foliation into the standard foliation given by\nthe Cartesian coordinates. Clearly, $\\Xi_n^{-1}(s,0)=\\gamma^u(s)$ (the\nunstable manifold of the origin parametrized by arc length and such that\n$\\gamma^u(0)=\\xi_n$), while $\\Xi_n^{-1}(0,s)=\\gamma_n^s(s)$ (the unstable\nmanifold of $\\xi_n$ parametrized by arc length). Finally we define\n$f_n:=(\\alpha_n\\beta_n)\\circ \\Xi_n$ with\n$\\alpha_n(z_1):=\\varsigma(\\operatorname{C}_5 n z_1)$, $\\beta_n(z_2):=\\varsigma(\\operatorname{C}_5n^{-1}\nz_2)$, for $C_5$ small enough, and \n\\[\n\\varsigma(x):=\\begin{cases}\n 1-|x+1|\\quad &|x+1|\\leq 1\\\\\n 0&|x+1|> 1.\n \\end{cases}\n\\]\nIn other words, $f_n$ is a function essentially supported on a\nneighborhood left of $\\xi_n$ of order $n^{-1}$ in the unstable direction\nand the stable. Accordingly, the supports of $f_n\\circ T^{-k}$ and $g$\nare disjoint for all $k\\leq \\operatorname{C} n$. Lemma \\ref{lem:holo} implies that\nthe the support is essentially a rhombus of size $n^{-1}$ and angle\n$n^{-1}$.\n\nThus we have\n\\[\n|g|_{\\Co^1}(\\|\\partial^u f_n\\|_{L^1(\\nu)}+\\|f_n\\|_{L^1(\\nu)})\\gamma_n\\geq\n\\int_{\\To^2}f_n\\int_{\\To^2}g\\geq \\operatorname{C} n^{-3}.\n\\]\nOn the other hand, using again Lemma \\ref{lem:holo},\n\\[\n\\|\\partial^u f_n\\|_{L^1(\\nu)}\\leq \\operatorname{C} n\\|(\\alpha'\\beta)\\circ\n\\Xi_n\\|_{L^1(\\nu)}\\leq \\operatorname{C} n^{-1}\n\\]\nwhich yields the Lemma.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:problems} Note that the norms in Lemma \\ref{lem:lower} and\nin Theorem \\ref{thm:main} (even in the stronger version given by\n\\eqref{eq:sharp}) are different. It is not obvious that, putting the\n$L^\\infty$ norm instead of the $L^1(\\nu)$ one keeps the same rate of\nmixing. More generally, it is well known that in the uniformly\nhyperbolic setting the smoothness of the function can have an influence\non the mixing rate. An analogous effect may arise in the present\nsetting but it remains to be investigate. A related problem that needs to be\naddressed is the higher dimensional analogous of the present model\nwhere the fixed point has different possibility of losing full\nhyperbolicty. It is clear that the present\nresult is only the starting point and not the end of the story.\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Correlated Gaussian environment\\label{SECgauss}}\n\nWe consider two beam splitters (with transmissivity $\\tau$) which combine\nmodes $A$ and $B$ with two environmental modes, $E_{1}$ and $E_{2}$,\nrespectively. These ancillary modes are in a zero-mean Gaussian state\n$\\rho_{E_{1}E_{2}}$ symmetric under $E_{1}$-$E_{2}$ permutation. In the\nmemoryless model, the environmental state is tensor-product $\\rho_{E_{1}E_{2\n}=\\rho\\otimes\\rho$, meaning that $E_{1}$ and $E_{2}$ are fully independent. In\nparticular, $\\rho$ is a thermal state with covariance matrix (CM)\n$\\omega\\mathbf{I}$, where the noise variance $\\omega=2\\bar{n}+1$ quantifies\nthe mean number of thermal photons $\\bar{n}$\\ entering the beam splitter. Each\ninteraction is then equivalent to a lossy channel with transmissivity $\\tau$\nand thermal noise $\\omega$.\n\n\\begin{figure}[ptbh]\n\\vspace{-2.8cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth] {duePICs.eps}\n\\end{center}\n\\par\n\\vspace{-2.3cm}\\caption{\\textit{Left}. Correlated Gaussian\nenvironment, with losses $\\tau$, thermal noise $\\omega$ and\ncorrelations $\\mathbf{G}$. The state of the environment $E_{1}$\nand $E_{2}$ can be separable or entangled. \\textit{Right}.\nCorrelation plane $(g,g^{\\prime})$ for the Gaussian environment,\ncorresponding to thermal noise $\\omega=2$. The black area\nidentifies forbidden environments (correlations are too strong to\nbe compatible with quantum mechanics). White area identifies\nphysical environments, i.e., the subset of points which satisfy\nthe bona-fide conditions of Eq.~(\\ref{CMconstraints}). Whitin this\narea, the inner region labbeled by S identifies separable\nenvironments, while the two outer regions identify entangled\nenvironments. Figures adapted from Ref.~\\cite{NJP} under a CC BY\n3.0 licence (http:\/\/creativecommons.org\/licenses\/by\/3.0\/).}\n\\label{ENVschemes}\n\\end{figure}\n\nThis Gaussian process can be generalized to include the presence of\ncorrelations between the environmental modes as depicted in the right panel of\nFig.~\\ref{ENVschemes}. The simplest extension of the model consists\\ of taking\nthe ancillary modes, $E_{1}$ and $E_{2}$, in a zero-mean Gaussian state\n$\\rho_{E_{1}E_{2}}$ with CM given by the symmetric normal form\n\\begin{equation}\n\\mathbf{V}_{E_{1}E_{2}}(\\omega,g,g^{\\prime})=\\left(\n\\begin{array}\n[c]{cc\n\\omega\\mathbf{I} & \\mathbf{G}\\\\\n\\mathbf{G} & \\omega\\mathbf{I\n\\end{array}\n\\right) ~, \\label{EVE_cmAPP\n\\end{equation}\nwhere $\\omega\\geq1$ is the thermal noise variance associated with each\nancilla, and the off-diagonal bloc\n\\begin{equation}\n\\mathbf{G=}\\left(\n\\begin{array}\n[c]{cc\ng & \\\\\n& g^{\\prime\n\\end{array}\n\\right) ~, \\label{Gblock\n\\end{equation}\naccounts for the correlations between the ancillas. This type of environment\ncan be separable or entangled (conditions for separability will be given afterwards).\n\nIt is clear that, when we consider the two interactions $A-E_{1}$ and\n$B-E_{2}$ separately, the environmental correlations are washed away. In fact,\nby tracing out $E_{2}$, we are left with mode $E_{1}$ in a thermal state\n($\\mathbf{V}_{E_{1}}=\\omega\\mathbf{I}$) which is combined with mode $A$ via\nthe beam-splitter. In other words, we have again a lossy channel with\ntransmissivity $\\tau$ and thermal noise $\\omega$. The scenario is identical\nfor the other mode $B$ when we trace out $E_{1}$. However, when we consider\nthe joint action of the two environmental modes, the correlation block\n$\\mathbf{G}$ comes into play and the global dynamics of the two travelling\nmodes becomes completely different from the standard memoryless scenario.\n\nBefore studying the system dynamics and the corresponding evolution of\nentanglement, we need to characterize the correlation block $\\mathbf{G}$\\ more\nprecisely. In fact, the two correlation parameters, $g$ and $g^{\\prime}$,\ncannot be completely arbitrary but must satisfy specific physical constraints.\nThese parameters must vary within ranges which make the CM of\nEq.~(\\ref{EVE_cmAPP}) a bona-fide quantum CM. Given an arbitrary value of the\nthermal noise $\\omega\\geq1$, the correlation parameters must satisfy the\nfollowing three bona-fide conditions~\\cite{TwomodePRA,NJP}\n\\begin{equation}\n|g|<\\omega,~~~|g^{\\prime}|<\\omega,~~~\\omega^{2}+gg^{\\prime}-1\\geq\n\\omega\\left\\vert g+g^{\\prime}\\right\\vert . \\label{CMconstraints\n\\end{equation}\n\n\n\\subsection{Separability properties}\n\nOnce we have clarified the bona-fide conditions for the environment, the next\nstep is to characterize its separability properties. For this aim, we compute\nthe smallest partially-transposed symplectic (PTS) eigenvalue $\\varepsilon$\nassociated with the CM\\ $\\mathbf{V}_{E_{1}E_{2}}$. For Gaussian states, this\neigenvalue represents an entanglement monotone which is equivalent to the\nlog-negativity~\\cite{logNEG1,logNEG2,logNEG3} $\\mathcal{E}=\\max\\left\\{\n0,-\\log\\varepsilon\\right\\} $. After simple algebra, we get~\\cite{NJP\n\\begin{equation}\n\\varepsilon=\\sqrt{\\omega^{2}-gg^{\\prime}-\\omega|g-g^{\\prime}|}~.\n\\end{equation}\nProvided that the conditions of Eq.~(\\ref{CMconstraints}) are satisfied, the\nseparability condition $\\varepsilon\\geq1$ is equivalent t\n\\begin{equation}\n\\omega^{2}-gg^{\\prime}-1\\geq\\omega|g-g^{\\prime}|~. \\label{sepCON\n\\end{equation}\n\n\nTo visualize the structure of the environment, we provide a numerical example\nin Fig.~\\ref{ENVschemes}. In the right panel of this figure, we consider the\n\\textit{correlation plane} which is spanned by the two parameters $g$ and\n$g^{\\prime}$. For a given value of the thermal noise $\\omega$, we identify the\nsubset of points which satisfy the bona-fide conditions of\nEq.~(\\ref{CMconstraints}). This subset corresponds to the white area in the\nfigure. Within this area, we then characterize the regions which correspond to\nseparable environments (area labelled by S) and entangled environments (areas\nlabelled by E).\n\n\\section{Direct distribution of entanglement in a correlated Gaussian\nenvironment\\label{SECdirect}}\n\nLet us study the system dynamics and the entanglement propagation in the\npresence of a correlated Gaussian environment, reviewing some key results from\nthe literature~\\cite{NJP}. Suppose that Charlie has an entanglement source\ndescribed by an EPR\\ state $\\rho_{AB}$ with C\n\\begin{equation}\n\\mathbf{V}(\\mu)=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\mu^{\\prime}\\mathbf{Z}\\\\\n\\mu^{\\prime}\\mathbf{Z} & \\mu\\mathbf{I\n\\end{array}\n\\right) ~, \\label{CM_TMSV\n\\end{equation}\nwhere $\\mu\\geq1$, $\\mu^{\\prime}:=\\sqrt{\\mu^{2}-1}$, and $\\mathbf{Z}$\\ is the\nreflection matri\n\\begin{equation}\n\\mathbf{Z}:=\\left(\n\\begin{array}\n[c]{cc\n1 & \\\\\n& -1\n\\end{array}\n\\right) ~. \\label{ZetaMAT\n\\end{equation}\nWe may consider the different scenarios depicted in the three panels of\nFig.~\\ref{twomodeEB}. Charlie may attempt to distribute entanglement to Alice\nand Bob as shown in Fig.~\\ref{twomodeEB}(1), or he may try to share\nentanglement with one of the remote parties, as shown in Figs.~\\ref{twomodeEB\n(2) and~(3).\n\n\\begin{figure}[ptbh]\n\\vspace{-0.2cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.60\\textwidth] {combined2.eps}\n\\end{center}\n\\par\n\\vspace{-0.9cm}\\caption{Scenarios for direct distribution of entanglement. (1)\nCharlie has two modes $A$ and $B$ prepared in an EPR state $\\rho_{AB}$. In\norder to distribute entanglement to the remote parties, Charlie transmits the\ntwo modes through the correlated Gaussian environment characterized by\ntransmissivity $\\tau$, thermal noise $\\omega$ and correlations $\\mathbf{G}$.\n(2) Charlie aims to share entanglement with Alice. He then keeps mode $B$\nwhile sending mode $A$ to Alice through the lossy channel $\\mathcal{E}_{A}$.\n(3) \\ Charlie aims to share entanglement with Bob. He then keeps mode $A$\nwhile sending mode $B$ to Bob through the lossy channel $\\mathcal{E}_{B}$.\n\\label{twomodeEB\n\\end{figure}\n\nLet us start considering the scenario where Charlie aims to share entanglement\nwith one of the remote parties (one-mode transmission). In particular, suppose\nthat Charlie wants to share entanglement with Bob (by symmetry the derivation\nis the same if we consider Alice). For sharing entanglement, Charlie keeps\nmode $A$ while sending mode $B$ to Bob as shown in Fig.~\\ref{twomodeEB}(3).\nThe action of the environment is therefore reduced to $\\mathcal{I}_{A\n\\otimes\\mathcal{E}_{B}$, where $\\mathcal{E}_{B}$ is a lossy channel applied to\nmode $B$. It is easy to check~\\cite{NJP} that the output state $\\rho\n_{AB^{\\prime}}$, shared by Charlie and Bob, is Gaussian with zero mean and C\n\\begin{equation}\n\\mathbf{V}_{AB^{\\prime}}=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\mu^{\\prime}\\sqrt{\\tau}\\mathbf{Z}\\\\\n\\mu^{\\prime}\\sqrt{\\tau}\\mathbf{Z} & x\\mathbf{I\n\\end{array}\n\\right) , \\label{ABpCM\n\\end{equation}\nwhere\n\\begin{equation}\nx:=\\tau\\mu+(1-\\tau)\\omega~.\n\\end{equation}\n\n\nRemarkably, we can compute closed analytical formulas in the limit of large\n$\\mu$, i.e., large input entanglement. In this case, the entanglement of the\noutput state $\\rho_{AB^{\\prime}}$ is quantified by the PTS\\ eigenvalu\n\\begin{equation}\n\\varepsilon=\\frac{1-\\tau}{1+\\tau}\\omega~.\n\\end{equation}\nThe EB condition corresponds to the separability condition $\\varepsilon\\geq1$,\nwhich provide\n\\begin{equation}\n\\omega\\geq\\frac{1+\\tau}{1-\\tau}:=\\omega_{\\text{EB}}~, \\label{EBcond\n\\end{equation}\nor equivalently $\\bar{n}\\geq\\tau\/(1-\\tau)$. Despite the EB condition of\nEq.~(\\ref{EBcond}) regards an EPR\\ input, it is valid for any input state. In\nother words, a lossy channel $\\mathcal{E}_{B}$\\ with transmissivity $\\tau$ and\nthermal noise $\\omega\\geq\\omega_{\\text{EB}}$ destroys the entanglement of any\ninput state $\\rho_{AB}$. Indeed Eq.~(\\ref{EBcond}) corresponds exactly to the\nwell-known EB condition for lossy channels~\\cite{HolevoEB}. The threshold\ncondition $\\omega=\\omega_{\\text{EB}}$ guarantees one-mode EB, i.e., the\nimpossibility for Charlie to share entanglement with the remote party.\n\nNow the central question is the following: Suppose that Charlie cannot share\nany entanglement with the remote parties (one-mode EB), can Charlie still\ndistribute entanglement to them? In other words, suppose that the correlated\nGaussian environment has transmissivity $\\tau$ and thermal noise\n$\\omega=\\omega_{\\text{EB}}$, so that the lossy channels $\\mathcal{E}_{A}$\\ and\n$\\mathcal{E}_{B}$ are EB. Is it still possible to use the joint channel\n$\\mathcal{E}_{AB}$ to distribute entanglement to Alice and Bob? In the\nfollowing, we explicitly reply to this question, discussing how entanglement\ncan be distributed by a separable environment, with the distributed amount\nbeing large enough to be distilled by one-way distillation\nprotocols~\\cite{NJP}.\n\nLet us study the general evolution of the two modes $A$ and $B$ under the\naction of the environment as in Fig.~\\ref{twomodeEB}(1). Since the input\nEPR\\ state $\\rho_{AB}$ is Gaussian and the environmental state $\\rho\n_{E_{1}E_{2}}$ is Gaussian, the output state $\\rho_{A^{\\prime}B^{\\prime}}$ is\nalso Gaussian. This state has zero mean and CM given by~\\cite{NJP\n\\begin{equation}\n\\mathbf{V}_{A^{\\prime}B^{\\prime}}=\\tau\\mathbf{V}_{AB}+(1-\\tau)\\mathbf{V\n_{E_{1}E_{2}}=\\left(\n\\begin{array}\n[c]{cc\nx\\mathbf{I} & \\mathbf{H}\\\\\n\\mathbf{H} & x\\mathbf{I\n\\end{array}\n\\right) ~,\n\\end{equation}\nwher\n\\begin{equation}\n\\mathbf{H}:=\\tau\\mu^{\\prime}\\mathbf{Z}+(1-\\tau)\\mathbf{G}~.\n\\end{equation}\nFor large $\\mu$, one can easily derive the symplectic spectrum of the output\nstat\n\\begin{equation}\n\\nu_{\\pm}=\\sqrt{\\left( 2\\omega+g^{\\prime}-g\\pm|g+g^{\\prime}|\\right)\n(1-\\tau)\\tau\\mu}~,\n\\end{equation}\nand its smallest PTS\\ eigenvalue~\\cite{NJP\n\\begin{equation}\n\\varepsilon=(1-\\tau)\\sqrt{(\\omega-g)(\\omega+g^{\\prime})}~, \\label{epsMAIN\n\\end{equation}\nquantifying the entanglement distributed to Alice and Bob.\n\nIn the same limit, one can compute the coherent\ninformation~\\cite{CohINFO,CohINFO2} $I(A\\rangle B)$ between the two remote\nparties, which provides a lower bound to the number of entanglement bits per\ncopy that can be distilled using one-way distillation protocols, i.e.,\nprotocols based on local operations and one-way classical communication. It is\nclear that one-way distillability implies two-way distillability, where both\nforward and backward communication is employed. After simple algebra, one\nachieves~\\cite{NJP\n\\begin{equation}\nI(A\\rangle B)=\\log\\frac{1}{e\\varepsilon}~. \\label{coheDIR\n\\end{equation}\nThus, remote entanglement is distributed for $\\varepsilon<1$ and is\ndistillable for $\\varepsilon1$).\nFurthermore, remote entanglement is present in the form of EPR correlations\nsince the two remote EPR quadratures $\\hat{q}_{-}^{r}:=(\\hat{q}_{a}-\\hat\n{q}_{b})\/\\sqrt{2}$ and $\\hat{p}_{+}^{r}:=(\\hat{p}_{a}+\\hat{p}_{b})\/\\sqrt{2}$\nhave variance\n\\begin{equation}\nV(\\hat{q}_{-}^{r})=V(\\hat{p}_{+}^{r})=\\mu^{-1}~.\n\\end{equation}\n\n\nThe simplest description of the entanglement swapping protocol can be given\nwhen we consider the limit for $\\mu\\rightarrow\\infty$. In this case the\ninitial states are ideal EPR states with quadratures perfectly correlated,\ni.e., $\\hat{q}_{a}=\\hat{q}_{A}$ and $\\hat{p}_{a}=-\\hat{p}_{A}$ for Alice, and\n$\\hat{q}_{b}=\\hat{q}_{B}$ and $\\hat{p}_{b}=-\\hat{p}_{B}$ for Bob. Then, the\noverall action of Charlie, i.e., the Bell measurement plus classical\ncommunication, corresponds to create a remote state with\n\\begin{equation}\n\\hat{q}_{b}=\\hat{q}_{a}-\\sqrt{2}q_{-},~\\hat{p}_{b}=-\\hat{p}_{a}-\\sqrt{2\np_{+}~.\n\\end{equation}\nThe quadratures of the two remote modes are perfectly correlated, up to an\nerasable displacement. In other words, the ideal EPR\\ correlations have been\nswapped from the initial states to the final conditional state $\\rho\n_{ab|\\gamma}$.\n\n\\subsection{Entanglement swapping in the presence of\ncorrelated-noise\\label{SECSUBswap2}}\n\nThe theory of entanglement swapping can be extended to include the presence of\nloss and correlated noise. We consider our model of correlated Gaussian\nenvironment with transmission $\\tau$, thermal noise $\\omega$ and correlations\n$\\mathbf{G}$. The modified scenario is depicted in Fig.~\\ref{swapLOSS\n.\\begin{figure}[ptbh]\n\\vspace{-1.5cm}\n\\par\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth] {swapLOSS.eps}\n\\end{center}\n\\par\n\\vspace{-1.9cm}\\caption{Entanglement swapping in the presence of loss, thermal\nnoise and environmental correlations (correlated Gaussian environment). The\nBell detector has been simplified.\n\\label{swapLOSS\n\\end{figure}\n\n\\subsubsection{Swapping of EPR\\ correlations}\n\nFor simplicity, we start by studying the evolution of the EPR correlations\nunder ideal input conditions ($\\mu\\rightarrow+\\infty$). After the classical\ncommunication of the outcome $\\gamma$, the quadratures of the remote modes $a$\nand $b$ satisfy the asymptotic relations\n\\begin{align}\n\\hat{q}_{b} & =\\hat{q}_{a}-\\sqrt{\\frac{2}{\\tau}}\\left( q_{-}-\\sqrt{1-\\tau\n}\\hat{\\delta}_{q}\\right) ,\\label{eqrrr}\\\\\n\\hat{p}_{b} & =-\\hat{p}_{a}-\\sqrt{\\frac{2}{\\tau}}\\left( p_{+}-\\sqrt{1-\\tau\n}\\hat{\\delta}_{p}\\right) , \\label{eqrrr2\n\\end{align}\nwhere $\\hat{\\delta}_{q}=(\\hat{q}_{E_{1}}-\\hat{q}_{E_{2}})\/\\sqrt{2}$ and\n$\\hat{\\delta}_{p}=(\\hat{p}_{E_{1}}+\\hat{p}_{E_{2}})\/\\sqrt{2}$ are noise\nvariables introduced by the environment.\n\nUsing previous Eqs.~(\\ref{eqrrr}) and~(\\ref{eqrrr2}), we construct the remote\nEPR\\ quadratures $\\hat{q}_{-}^{r}$\\ and $\\hat{p}_{+}^{r}$, and we compute the\nEPR variance\n\\begin{equation}\n\\boldsymbol{\\Lambda}:=\\left(\n\\begin{array}\n[c]{cc\nV(\\hat{q}_{-}^{r}) & \\\\\n& V(\\hat{p}_{+}^{r})\n\\end{array}\n\\right) \\rightarrow\\boldsymbol{\\Lambda}_{\\infty}=\\frac{1-\\tau}{\\tau\n(\\omega\\mathbf{I}-\\mathbf{ZG})~, \\label{lam1\n\\end{equation}\nwhere the limit is taken for $\\mu\\rightarrow+\\infty$. Assuming the EB\ncondition $\\omega=\\omega_{\\text{EB}}$, we finally ge\n\\begin{equation}\n\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}=\\frac{1}{\\tau}\\left[ (1+\\tau\n)\\mathbf{I}-(1-\\tau)\\mathbf{ZG}\\right] . \\label{lam2\n\\end{equation}\n\n\nIn the case of a memoryless environment ($\\mathbf{G=0}$) we see that\n$\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}=(1+\\tau^{-1})\\mathbf{I}\\geq\n\\mathbf{I}$, which means that the EPR\\ correlations cannot be swapped to the\nremote systems. However, it is evident from Eq.~(\\ref{lam2}) that there are\nchoices for the correlation block $\\mathbf{G}$\\ such that the EPR\\ condition\n$\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}<\\mathbf{I}$ is satisfied. For\ninstance, this happens when we consider $\\mathbf{G=}g\\mathbf{Z}$. In this case\nit is easy to check that $\\boldsymbol{\\Lambda}_{\\infty,\\text{EB}}<\\mathbf{I}$\nis satisfied for $\\tau\\geq1\/4$ and $g>(1-\\tau)^{-1}$. Under these conditions,\nEPR\\ correlations are successfully swapped to the remote modes. In particular,\nfor $\\tau>1\/2$ and $g>(1-\\tau)^{-1}$ there are separable environments which do\nthe job.\n\n\\subsubsection{Swapping and distillation of entanglement}\n\nHere we discuss in detail how entanglement is distributed by the swapping\nprotocol in the presence of a correlated Gaussian environment. In particular,\nsuppose that Alice and Bob cannot share entanglement with Charlie because the\nenvironment is one-mode EB. Then, we aim to address the following questions:\n(i)~Is it still possible for Charlie to distribute entanglement to the remote\nparties thanks to the environmental correlations? (ii)~In particular, is the\nswapping successful when the environmental correlations are separable?\n(iii)~Finally, are Alice and Bob able to distill the swapped entanglement by\nmeans of one-way distillation protocols? Our previous discussion on EPR\ncorrelations clearly suggests that these questions have positive answers. Here\nwe explicitly show this is indeed true for quantum entanglement by finding the\ntypical regimes of parameters that the Gaussian environment must satisfy.\n\nIn order to study the propagation of entanglement we first need to derive the\nCM\\ $\\mathbf{V}_{ab|\\gamma}$\\ of the conditional remote state $\\rho\n_{ab|\\gamma}$. As before, we have two identical EPR\\ states at Alice's and\nBob's stations with CM $\\mathbf{V}(\\mu)$ given in Eq.~(\\ref{CM_TMSV}). The\ntravelling modes $A$ and $B$ are sent to Charlie through a Gaussian\nenvironment with transmissivity $\\tau$, thermal noise $\\omega$ and\ncorrelations $\\mathbf{G}$. After the Bell measurement and the classical\ncommunication of the result $\\gamma$, the conditional remote state at Alice's\nand Bob's stations is Gaussian with CM~\\cite{BellFORMULA\n\\begin{equation}\n\\mathbf{V}_{ab|\\gamma}=\\left(\n\\begin{array}\n[c]{cc\n\\mu\\mathbf{I} & \\\\\n& \\mu\\mathbf{I\n\\end{array}\n\\right) -\\frac{(\\mu^{2}-1)\\tau}{2}\\left(\n\\begin{array}\n[c]{cccc\n\\frac{1}{\\theta} & & -\\frac{1}{\\theta} & \\\\\n& \\frac{1}{\\theta^{\\prime}} & & \\frac{1}{\\theta^{\\prime}}\\\\\n-\\frac{1}{\\theta} & & \\frac{1}{\\theta} & \\\\\n& \\frac{1}{\\theta^{\\prime}} & & \\frac{1}{\\theta^{\\prime}\n\\end{array}\n\\right) ~, \\label{VabGamma\n\\end{equation}\nwher\n\\begin{equation}\n\\theta=\\tau\\mu+(1-\\tau)(\\omega-g),~\\theta^{\\prime}=\\tau\\mu+(1-\\tau\n)(\\omega+g^{\\prime})~. \\label{thetas\n\\end{equation}\n\n\nFrom the CM of Eq.~(\\ref{VabGamma}) we compute the smallest PTS eigenvalue\n$\\varepsilon$ quantifying the remote entanglement at Alice's and Bob's\nstations. For large input entanglement $\\mu\\gg1$, we find a closed formula in\nterms of the environmental parameters, i.e.\n\\begin{equation}\n\\varepsilon=\\frac{1-\\tau}{\\tau}\\sqrt{(\\omega-g)(\\omega+g^{\\prime\n)}:=\\varepsilon(\\tau,\\omega,g,g^{\\prime})~, \\label{SpectrumTOT\n\\end{equation}\nwhich is equal to Eq.~(\\ref{epsMAIN}) up to a factor $\\tau^{-1}$. As before,\nthis eigenvalue not only determines the log-negativity but also the coherent\ninformation $I(a\\rangle b)$\\ associated with the remote state $\\rho\n_{ab|\\gamma}$. In fact, for large $\\mu$, one can easily compute the asymptotic\nexpressio\n\\begin{equation}\nI(a\\rangle b)\\rightarrow\\log\\frac{2}{e}\\sqrt{\\frac{\\det\\mathbf{V}_{b|\\gamma\n}{\\det\\mathbf{V}_{ab|\\gamma}}}=\\log\\frac{1}{e\\varepsilon}~,\n\\label{IabCOHEswap\n\\end{equation}\nwhich is identical to the formula of Eq.~(\\ref{coheDIR}) for the case of\ndirect distribution. Thus, the PTS\\ eigenvalue of Eq.~(\\ref{SpectrumTOT})\ncontains all the information about the distribution and distillation of\nentanglement in the swapping scenario. For $\\varepsilon<1$ entanglement is\nsuccessfully distributed by the swapping protocol (log-negativity\n$\\mathcal{E}>0$). Then, for the stronger condition $\\varepsilon1\/2$. In this regime, in fact, the distribution of remote entanglement\ncan be activated by separable environments. As explicitly shown for\n$\\tau=0.75$ and $0.9$, the activation area progressively invades the region of\nseparable environments. In other words, separable correlations become more and\nmore important for increasing transmissivities. Furthermore, for $\\tau\n\\gtrsim0.75$, separable environments are even able to activate the\ndistribution of distillable entanglement ($\\varepsilon, green] (x1) -- (y);\n\t\t\\draw[->, green] (x2) -- (y);\n\t\t\\draw[->, blue] (x3) -- (x2);\n \\end{tikzpicture}\n \\caption{Directed Acyclic Graph (DAG) illustrating the data generating process.}\n \\label{fig:example-intro}\n\\end{figure}\n\\noindent Direct importance measures such as Permutation Feature Importance (PFI) \\cite{Breiman2001rf} or marginal SAGE \\cite{covert_understanding_2020} consider the feature \\textit{race} irrelevant, as it is not directly causal for the model's predictions.\nAs a result, they fail to expose the leakage of information from \\textit{ethnicity} via the proxy variable \\textit{zip code}.\nMeasures of associative importance such as conditional SAGE \\cite{covert_understanding_2020}\nwould expose the association and consider \\textit{ethnicity} to be relevant.\nHowever, even in combination with direct importance, associative importance does not provide insight into via which of the directly important features (\\textit{zip code} or \\textit{job experience}) the protected variable leaks into the model. Consequently we are not guided towards sparse interventions that eliminate the association from the model.\\\\\nWe propose a framework (\\underline{De}composition of Global Feature Importance into \\underline{D}irect and \\underline{A}ssociative \\underline{C}omponen\\underline{t}s, DEDACT) that enables a holistic understanding of both sources of prediction-relevant information \\textit{and} feature pathways which allow the respective sources of information to enter the model's predictions. With DEDACT one can either decompose the direct importance of a feature into the contributions of its sources of information, or one can decompose the associative importance of an unwanted influence into its direct feature pathway components.\\\\\nDEDACT does not require knowledge of the underlying dependence structure or causal graph. As is the case with associative importance measures such as conditional SAGE \\cite{covert_understanding_2020}, DEDACT requires sampling from conditional distributions.\nThese samples can be reused for the decomposition; in this way the computational overhead can be reduced.\\\\\nBefore the decomposition will be introduced in Section \\ref{sec:decomposition}, we will formally define associative and direct importance in Section \\ref{sec:disentangling-direct-and-indirect} and measures for the respective direct and associative components in Section \\ref{sec:iai-via-and-di-from}. The method will be validated on simulated data in Section \\ref{sec:application}.\n\n\\subsection{Related Work and Contributions}\nPrevious research on local SHAP based feature effect explanations have studied direct and indirect effects. Heskes et al. \\cite{heskes_causal_2020} decompose causal Shapley value functions into one direct and one indirect component. Our method is applicable to a wide range of global direct and associative feature importance methods and allows their full decomposition into feature components. Wang et al. \\cite{wang_shapley_2020} offer a graph-based decomposition of direct and indirect causal effects for local causal SHAP explanations. However, their approach (1) cannot be applied to direct or associative importance methods, or (2) to feature importance methods and (3) requires knowledge of the underlying causal graph. Relative Feature Importance \\cite{konig_relative_2021} quantifies the direct importance of a variable that cannot be attributed to a user specified set of variables. However, the method has not been applied to decompose the importance of a feature into its prediction-relevant sources of information. Moreover, it does not allow to decompose the associative importance of a variable. Our contribution is the first decomposition of global direct and associative importance measures into their respective associative or direct components, and therefore provides novel insight into both model and data.\\\\\n\\section{Notation}\n\\label{sec:background-and-notation}\nThe Table in Figure \\ref{fig:notation} summarizes the mathematical notation and terms used throughout the article.\nIn order to formalize a distinction between direct and associative importance we differentiate between variables on the data level and features on the model level. We refer to the real-world concept as variable $X_j$ and to the corresponding model input as feature $\\underline{X}_j$ (Figure \\ref{fig:notation}, left panel). Although in the standard setting variable and feature take the same values, they are conceptually different. If we say that we intervene on a feature, it only means that we change the input to the respective feature. If we say that a feature is not causal for the prediction, the corresponding real world concept may still be causal since it could indirectly affect the prediction via downstream variables. Following \\cite{janzing2020feature}, we denote perturbations of the features using the do-operator \\cite{pearl2009causality}, i.e. the term $do(\\underline{X}_k = \\tilde{X}_k)$ denotes the replacement of the input $\\underline{X}_k$ with the perturbed variable $\\tilde{X}_k$. The corresponding prediction is denoted as $\\hat{Y}|{do(\\underline{X}_k = \\tilde{X}_k)}$.\\\\\nSome methods rely on marginalized prediction functions. To be more specific, the prediction function $f$ that is marginalized over $\\underline{X}_S$ is defined as\n$$ f_S(\\underline{x}_{S}) = \\mathbb{E}_{\\underline{X}_{D \\backslash S}|\\underline{X}_S}[f(\\underline{X}_{D \\backslash S}) | \\underline{X}_S = \\underline{x}_{S}],$$\nwhere the conditional expectation is taken over the feature distribution. If features are sampled independently this term reduces to the marginal expectation.\\\\\nWe indicate whether the prediction of the original model $f$ or the marginalized variant $f_S$ is meant by making use of the respective subscripts: $f$ or $f_S$. For example, $\\hat{Y}_{f_S}$ refers to the marginalized prediction.\\\\\nThere will be frequent references to variables as \"containing information\" or to \"information entering a variable\". Such phrases are not meant in the information theoretic sense \u2013 despite links with concepts such as mutual information \u2013 but rather figuratively to enable a more pleasant reading experience.\n\n\\input{figures\/figure-notation}\n\n\\section{Background: Direct and Associative Importance}\n\\label{sec:disentangling-direct-and-indirect}\nOver the course of this Section, we formally introduce general definitions of direct and associative importance that generalize methods such as Permutation Feature Importance \\cite{Breiman2001rf} and SAGE value functions \\cite{covert_understanding_2020}. These two classes of relevance, namely direct and associative importance, are introduced in Sections \\ref{subsec:dr} and \\ref{subsec:ar}. As the proposed decomposition is based on these general definitions, it is applicable to a wide-range of feature importance methods.\\\\\nBoth definitions can be modified along two axes: Firstly, recent proposals \\cite{Datta2017,lundberg_unified_2017,covert_understanding_2020} formulate feature relevance quantification as a cooperative game, which evaluates the relevance of features\/variables with respect to different perturbation baselines.\\footnote{Feature relevance attribution can be seen as a cooperative game, where the contribution of one player (feature\/variable) depends on which players are already in the room \\cite{lundberg_unified_2017,covert_understanding_2020}. For example, if two features contribute to the prediction performance via an interaction term, their joint contribution is attributed to the feature that is perturbed first. If the relevance of features is only considered in isolation, both features are attributed with the interaction contribution. Consequently, recent methods do not only evaluate the relevance in isolation but also over multiple different baselines \\cite{covert_understanding_2020}.} In order to accommodate such considerations within our framework, the general definitions allow the specification of flexible perturbation baselines (more details below). Furthermore, the general measures accommodate methods that rely on the original prediction function $f$, such as Permutation Feature Importance (PFI) \\cite{Breiman2001rf} as well as methods that marginalize the prediction function over perturbed features, as suggested by \\cite{covert_understanding_2020}.\\\\\n\\input{figures\/dag-intuition-di-tai}\n\\subsection{Direct Importance}\n\\label{subsec:dr}\n\nDirect importance measures assess the relevance of specific features for the model's performance. They are based on a simple idea: If a feature is not important, removing all its prediction-relevant information should have no or only little effect on the model's performance.\n\\\\\nAs such, direct importance measures compare the model performance before and after perturbing the features of interest. The perturbed variable $\\tilde{X}_K^\\emptyset$ preserves the marginal distribution of the original variable, but is sampled independently from $X$ and $Y$ (as indicated by the superscript $\\emptyset$). Hence, the perturbed variable only contains noise and no prediction-relevant information.\\\\\nOur notation starts off with the full perturbation of all features. After that, we quantify the improvement in performance when restoring a set of features. That is, the direct importance $DI(\\underline{X}_K)$ quantifies the improvement in performance when features $\\underline{X}_K$ are restored in isolation. The direct importance $DI(\\underline{X}_K|\\underline{X}_B)$ over baseline $B$ quantifies the improvement in performance, when $\\underline{X}_K$ is restored in a context where features $\\underline{X}_B$ have been restored already. Furthermore, we indicate whether the original model $f$ or the marginalized prediction functions $f_{B}, f_{B \\cup K}$ are used by $DI(\\underline{X}_K|\\underline{X}_B; f, f)$ and $DI(\\underline{X}_K|\\underline{X}_B; f_{B}, f_{B \\cup K})$. For example, Permutation Feature Importance is a special case of a direct importance measure, i.e. $DI(\\underline{X}_K|\\underline{X}_{D \\backslash K}; f, f)$. Direct importance is formally introduced in Definition \\ref{def:dr} and illustrated in Figure \\ref{fig:dag-intuition-dr-ar} (a) - (b).\\\\\n\\begin{definition}[Direct Importance (DI)]\nGiven two disjoint feature index sets $B$ and $K$, direct importance of features $\\underline{X}_K$ over baseline features $\\underline{X}_B$ is defined as:\n\\begin{align*}\n DI(\\underline{X}_K|\\underline{X}_B;g,g') := &\\mathbb{E}[\\mathcal{L}(\\hat{Y}_{g}|{do(\\underline{X}_{D \\backslash B }=\\tilde{X}_{D \\backslash B}^\\emptyset)};Y)]\\\\ \n - &\\mathbb{E}[\\mathcal{L}(\\hat{Y}_{g'}|{do(\\underline{X}_{D \\backslash (B \\cup K)} = \\tilde{X}_{D \\backslash (B \\cup K)}^\\emptyset)};Y)]\n\\end{align*}\nwhere $(g, g') \\in \\{(f, f),(f_{B}, f_{(B \\cup K)})\\}$. In addition, the perturbed joint $\\tilde{X}^{\\emptyset}$ preserves the covariate joint distribution, but is independent of the original variables and the prediction target. I.e., $P(\\tilde{X}^{\\emptyset}) = P(X) \\text{ and }\\tilde{X}^{\\emptyset} \\Perp (X, Y)$.\n\\label{def:dr}\n\\end{definition}\nAs all features except the features of interest are left unchanged, direct importance can only enter the model via the features of interest themselves. Therefore, directly important features must be causal for the prediction (Proposition \\ref{proposition:tdr-direct-sensitivity}).\\\\\n\\begin{proposition}[Direct Sensitivity of DI]\nNonzero Direct Importance $DI(\\underline{X}_K|\\underline{X}_B)$ implies that feature $\\underline{X}_K$ is causal for the model (over $P(X_B)P(X_K)P(X_{D \\backslash (B \\cup K)})$).\n\\label{proposition:tdr-direct-sensitivity}\n\\end{proposition}\nHowever, as illustrated by the motivational example in the introduction to this paper, direct importance fails to expose leakage of information from variables that are not directly used by the model.\n\\subsection{Associative Importance}\n\\label{subsec:ar}\nIn contrast to direct importance, associative importance measures are not concerned with the mechanisms that the model uses. They rather aim at measuring the component of the predictive performance that can be explained with information contained in the variables of interest $X_J$.\\\\\nTherefore, associative importance measures compare the model's performance under fully perturbed features with the performance that results from reconstruction of the features by the use of information from the variables of interest $X_J$. The reconstructed features preserve the conditional distribution of their corresponding variables with the variable of interest, i.e. $P(\\underline{X}|X_J) = P(X|X_J)$. Consequently, the features corresponding to the variables of interest $\\underline{X}_J$ are fully reconstructed, features that correspond to dependent variables are partly reconstructed, and features that correspond to variables that are independent from $X_J$ remain unchanged.\\\\\nAs for direct importance, it is of relevance whether the variable is introduced in isolation, or whether it is added to a set of variables for which the information has been restored already. We denote the associative importance of variables $X_J$ that are introduced in isolation as $AI(X_J)$, and the associative importance of variables $X_J$ that are introduced to already conditioned upon variables $X_C$ as $AI(X_J|X_C)$. Associative importance is formally introduced in Definition \\ref{def:ar} and illustrated in Figure \\ref{fig:dag-intuition-dr-ar} (c) - (d).\n\\begin{definition}[Associative Importance (AI)]\nWe define the associative importance of a variable set $J$ given context variables $C$ as:\n\\begin{align*}\n AI(X_J|X_C; g, g') := &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_g|{do(\\underline{X}_{D\\backslash C} = \\tilde{X}_{D\\backslash C}^{C})})]\\\\\n- &\\mathbb{E}[\\mathcal{L}(Y;\\hat{Y}_{g'}|{do(\\underline{X}_{D\\backslash (C \\cup J)} = \\tilde{X}_{D \\backslash (C \\cup J)}^{C \\cup J})})]\n\\end{align*}\nwhere $(g,g') \\in \\{(f,f), (f_{C},f_{C \\cup J})\\}$.\nThe conditionally perturbed joint $\\tilde{X}^{S}$ is required to preserve the covariate joint distribution as well as the conditional distribution with features $X_S$, i.e. $P(\\tilde{X}^{S}| X_S) = P(X|X_S)$. Also, it is conditionally independent of the remaining covariates and prediction target ($\\tilde{X}^{S} \\Perp (X_{D \\backslash S}, Y) | X_S$).\n\\label{def:ar}\n\\end{definition}\nAssociative importance provides insight into the dependence structure in the underlying dataset. Nonzero associative importance indicates that $X_J$ contains prediction-relevant information that is not included in the context variables $X_C$.\n\\begin{proposition} [Associative Sensitivity of AI]\nNonzero associative importance $AI(X_J|X_C;f,f)$ based on the non-marginalized function $f$ implies that the target $Y$ is conditionally dependent with $X_J$ given $X_C$, i.e. $Y \\not \\idp X_J | X_C$. If the marginalized functions $f_C$ and $f_{C \\cup J}$ are used, nonzero associative importance $AI(X_J|X_C;f_C, f_{C \\cup J})$ implies conditional dependence of $Y$ with $X_J$ given $X_C$, i.e. $Y \\not \\idp X_J | X_C$ if (1) cross entropy or mean squared error is used as loss and (2) $f$ is the respective loss-optimal predictor.\n\\label{proposition:tar-associative-sensitivity}\n\\end{proposition}\nYet, for the computation of associative importance not only the distribution of the features $\\underline{X}_J$ that correspond to the variables of interest $X_J$ but also all the features corresponding to dependent variables of $X_J$ is affected. Associative importance therefore enters the model directly and\/or indirectly. As a result, it fails to provide insight into which features the model uses for its prediction. An illustrative example for the lacking causal interpretation of associative importance is given in Section \\ref{sec:introduction}.\\\\\nWith the right choice of parameters, the definitions of associative importance and direct importance yield well-established IML methods. An overview is given in Table \\ref{table:influence-family-members}. In order to estimate the measures based on real data, the expected loss (risk) can be replaced by the empirical risk.\\\\\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{l|l|l|l}\n\\textbf{type} & \\textbf{$(g,g')$} & \\textbf{sets} & \\textbf{method}\\\\\n\\midrule\n\\textbf{DI} & $(f, f)$ & $K = \\{k\\}$ $B = D \\backslash K$, & Permutation Feature Importance of $k$ \\cite{Breiman2001rf}\\\\\n\n & $(f_{\\emptyset}, f_{S})$ & $K = S$, $B = \\emptyset$ & marginal SAGE $v(S)$ \\cite{covert_understanding_2020} \\\\\n\n\\midrule\n\\textbf{AI} & $(f_{C}, f_{C \\cup J})$ & $J = S$, $C = \\emptyset$ & conditional SAGE $v(S)$ \\cite{covert_understanding_2020}\\\\\n\n & $(f, f)$ & $J=\\{j\\}$, $C = D \\backslash J$ & Conditional Feature Importance of $j$ \\cite{Strobl2008}\\\\\n\\end{tabular}\n\\caption{Parameter choices for direct importance (DI) and associative importance (AI) that yield well-established IML methods.}\n\\label{table:influence-family-members}\n\\end{table}\n\\section{Components of Associative and Direct Importance}\n\\label{sec:iai-via-and-di-from}\nIn Section \\ref{sec:disentangling-direct-and-indirect} classes of direct and associative importance measures have been formally introduced. We have shown that both measures have strengths and weaknesses: While direct importance measures allow the exposure of those features that causally influence the prediction performance, they fail to uncover leakage of information from variables that the model does not directly use for its prediction. In contrast to that, associative importance measures can uncover leakage of information from variables of interest, irrespective of whether the corresponding feature is causal for the model's prediction and performance. However, as opposed to direct importance, associative importance measures do not incline a causal interpretation: If the associative importance for a variable is nonzero, we cannot infer that the corresponding feature is causal for the prediction. Even in combination, the methods fail to provide insight into both the sources of prediction-relevant information and the feature pathways, which allow the information to enter the model.\\\\\nIn order to mitigate this inherent trade-off, we propose two measures. Firstly, \\textit{DI from} that quantifies the component of the direct importance of a feature of interest $\\underline{X}_K$ that can be explained with information from a set of variables $X_J$. Using \\textit{DI from}, we get insight into the sources of information that enter the model via a feature $\\underline{X}_K$. Secondly, \\textit{AI via} that quantifies the component of the associative importance of a variable $X_J$ that enters the model via a specific set of features $\\underline{X}_K$. Using \\textit{AI via}, we learn which feature pathways allow the information of variables $X_J$ to enter the model.\\\\\n\\input{figures\/dag-intuition-tai-via-di-from}\n\\subsection{Components of Direct Importance}\nWhen computing the direct importance of a feature, we compare the prediction when $\\underline{X}_K$ is fully perturbed with the prediction when $\\underline{X}_K$ is fully reconstructed. Now, we would like to understand whether $X_J$ contributes to the direct importance of $\\underline{X}_K$. Therefore, instead of fully reconstructing $\\underline{X}_K$ we only partially reconstruct the feature using information from $X_J$. As such, we quantify the degree to which the direct importance of $\\underline{X}_K$ can be explained with $X_J$. An illustration of this procedure is given in Figure \\ref{fig:dag-intuition-tai-via-di-from} (a)-(b).\\\\\n\\begin{definition}[DI from]\nFor the direct importance of features $K$ given baseline $B$, we define the associative component that can be attributed to variables $J$ as \n\\begin{align*}\n\tDI(\\underline{X}_K|\\underline{X}_B \\leftarrow X_J;g, g')\n\t:= &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_g|{do(\\underline{X}_{D \\backslash B} = \\tilde{X}_{D \\backslash B}^{\\emptyset})})] \\\\\n\t- &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g'}|{do(\\underline{X}_{D \\backslash B} = \\tilde{X}_{D \\backslash B}^{\\emptyset}, \\underline{X}_{K} = \\tilde{X}_{K}^{J})})]\n\\end{align*}\nwhere $(g, g') \\in \\{(f, f),(f_{B}, f_{(B \\cup K)})\\}$. In addition, $\\tilde{X}^\\emptyset$ and $\\tilde{X}^J$ satisfy the following requirements: $P(\\tilde{X}^\\emptyset) = P(X)$ and $\\tilde{X}^\\emptyset \\Perp X$ as well as $P(\\tilde{X}^J|X_J) = P(X|X_J)$ and $\\tilde{X}^J \\Perp X | X_J$.\n\\label{def:tdi-from}\n\\end{definition}\nIntuitively, if the variables $X_J$ do not contain information, the respective features $\\underline{X}_K$ cannot be reconstructed at all. Consequently, the direct importance $DI(\\underline{X}_K|\\underline{X}_B)$ cannot be attributed to $X_J$. In contrast, if $X_J$ were to be perfectly correlated with $\\underline{X}_K$, the feature is fully reconstructed and the full relevance can be attributed to $X_J$.\\\\\nIn scenarios where all features except for the features of interest are reconstructed in the baseline $B = D \\backslash K$, \\textit{DI from} is complementary to Relative Feature Importance \\cite{konig_relative_2021}, which quantifies the component of Permutation Feature Importance that cannot be attributed to a user-defined set of variables $X_G$.\\\\\n \n\\subsection{Components of Associative Importance}\n\nWhen computing the associative importance $AI(X_J|X_C)$, we compare the prediction when \\textit{all} features are reconstructed using $X_C$ with the prediction when \\textit{all} features are reconstructed using $X_C$ and $X_J$. In order to gain insight into the direct feature contributions to the associative importance, we want to quantify how much specific feature pathways $\\underline{X}_K$ contribute to the overall score. Therefore, instead of reconstructing the information from $X_J$ in all features, we only update the features of interest $\\underline{X}_K$. In other words, we allow information from $X_J$ over $X_C$ to enter the model via $\\underline{X}_K$ while blocking all other feature paths. An illustration is provided in Figure \\ref{fig:dag-intuition-tai-via-di-from} (c)-(d).\\\\\n\\begin{definition}[AI via]\nThe Associative Importance of variable set $X_J$ given context variables $X_C$ via features $\\underline{X}_K$ is defined as\n\\begin{align*}\n\tAI(X_J|X_C \\rightarrow \\underline{X}_K;g,g') \n\t= &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g}|{do(\\underline{X}_D = \\tilde{X}_D^{C})})] \\\\\n\t- &\\mathbb{E}[\\mathcal{L}(Y; \\hat{Y}_{g'}|{do(\\underline{X}_{K} = \\tilde{X}_{K}^{C \\cup J}, \\underline{X}_{D \\backslash K} = \\tilde{X}_{D \\backslash K}^{C})})]\\\\\n\t= &AI(X_J|X_C;g,g') | {do(\\underline{X}_{D \\backslash K} = \\tilde{X}_{D \\backslash K}^{C})}\n\\end{align*}\nfor $(g,g') \\in \\{(f,f), (f_{C},f_{C \\cup J})\\}$. Furthermore, $\\tilde{X}^C$ and $\\tilde{X}^{C \\cup J}$ satisfy the following requirements: $P(\\tilde{X}^C | X_C) = P(X | X_C)$ and $\\tilde{X}^C \\Perp X | X_C$ as well as $P(\\tilde{X}^{C \\cup J} | X_{C \\cup J}) = P(X | X_{C \\cup J})$ and $\\tilde{X}^{C \\cup J} \\Perp X | X_{C \\cup J}$.\n\\label{def:tai-via}\n\\end{definition}\nIntuitively, \\textit{AI via} can be thought of as the direct importance of a feature of interest, except that instead of fully perturbing the features only surplus information w.r.t to $X_C$ is removed, and that features are only reconstructed with respect to $X_C$ and $X_J$ instead of fully reconstructing them.\\\\\n\\\\\nIt should be noted, that both measures are based on model evaluations on mixed joint distributions of the form $\\tilde{X}_{K}^{C \\cup J}, \\tilde{X}_{D \\backslash K}^{C}$. As the variables are being defined independently, the components are only linked via the conditioning variable $C$. Consequently, if $X_K$ and $X_{D \\backslash K}$ are dependent conditional on $X_C$, the mixed joint distribution does not preserve the original covariate joint. Such extrapolation may cause misleading interpretations \\cite{hooker_please_2019,molnar2020pitfalls}. We conjecture that coupling the variables, e.g. with conditional-rank preserving independent representation learning, can avoid unnecessary extrapolation \\cite{lum_statistical_2016,johndrow_algorithm_2019,Zhao2020Conditional}. However, this investigation goes beyond the scope of the article.\n\n\\section{Decomposing Associative and Direct Importance with DEDACT}\n\\label{sec:decomposition}\n\nIn the previous Section, we have defined two measures: \\textit{DI from} that quantifies the component of a direct importance that can be attributed to a set of variables $X_J$. And \\textit{AI via} that quantifies the component of the associative importance that enters the model via a set of features $\\underline{X}_K$.\\\\\nWe can leverage these measures in various ways to yield a decomposition of associative and direct importance. The choice of decomposition depends on the prerequisites we place on the method: For scenarios where computational efficiency is prioritized, we introduce a fast decomposition in Section \\ref{subsec:fast-decomposition}. For situations where computational resources are available and a fair attribution is prioritized, we propose an additive Shapley value based decomposition in Section \\ref{subsec:shapley-decomposition}.\n\\begin{figure}[H]\n\\centering\n \\input{figures\/tikz-intuition-attribution}\n \\caption{Notation for decompositions of direct and associative importance. (a) The associative importance of a variable (denoted as $\\alpha_j$) is decomposed into its feature contributions (denoted as $\\alpha_{jk}$). (b) The direct importance of a feature (denoted as $\\beta_k$) is decomposed into its variable contributions (denoted as $\\beta_{kj}$). The feature\/variable of interest is highlighted in orange, the respective direct and indirect components are highlighted in green and blue.}\n \\label{fig:my_label}\n\\end{figure}\n\\subsection{DEDACT: Fast Decomposition}\n\\label{subsec:fast-decomposition}\n\nFor the purpose of computational efficiency, we restrict the fast decomposition to require the evaluation of only one further \\textit{AI via}\/\\textit{DI from} configuration per evaluation of $AI$\/$DI$ and component. Furthermore, we prioritize sensitivity, referred to the exposure of sources of information and feature pathways, over additivity. One direct and one associative importance measure will be considered as examples.\\\\\n\\\\\n\\textbf{Permutation Feature Importance:} In our notation, Permutation Feature Importance \\cite{Breiman2001rf} is written as:\n$$\\beta_k := PFI_k = DI(\\underline{X}_k | \\underline{X}_{D \\backslash k}; f, f)$$\nUsing a single evaluation of \\textit{DI from}, we can quantify the component of the feature importance that can be explained by a specific variable $X_j$.\n$$\\beta_{kj} := DI(\\underline{X}_k | \\underline{X}_{D \\backslash k} \\leftarrow X_j; f, f)$$\nThe component $\\beta_{kj}$ represents the total contribution that the variable $X_j$ is able to provide in isolation. Nevertheless, the contribution that $X_j$ can only deliver in cooperation with a further feature $X_l$ is not quantified. Further, if variables $X_j$ and $X_l$ are dependent, then both are fully attributed with the shared contribution. Consequently, the attributions are not guaranteed to add up to the overall importance $\\beta_k$.\\\\\nIn order to achieve an \\textit{additive} decomposition with just one evaluation per feature and component, we would need to define an order of the variables $\\pi$. This estimate would then be\n$$\\beta_{kj}^{\\pi} = DI(\\underline{X}_K | \\underline{X}_{D \\backslash k} \\leftarrow X_{S \\cup j}) - DI(\\underline{X}_K | \\underline{X}_{D \\backslash k} \\leftarrow X_S),$$\nwhere $S$ is the set of features that appears earlier in the order $\\pi$. Although such an order based decomposition is additive, it has a major disadvantage: The attribution of relevance to the variables is strictly prioritized. Variables that appear earlier in the order are fully attributed with the contribution that they share with variables that appear later in the order. As such, features that contain prediction-relevant information that enters via $\\underline{X}_k$ but that share their full contribution with variables that appear earlier in the order, are deemed irrelevant. Additionally, the contribution that two variables jointly achieve is fully attributed to the variable that appears later in the order.\nConsequently, since sensitivity is prioritized, $\\beta_{kj}$ is to be preferred over $\\beta_{kj}^{\\pi}$.\\\\\n\\\\\n\\textbf{Conditional SAGE:} SAGE \\cite{covert_understanding_2020} is based on a linear combination of value functions $v$. In order to compute the surplus contribution of variable $j$ given the set of features $C$, the difference in value with and without the respective variable is computed. In our notation, these differences can be written as\n$$\\alpha_j^C = v(C \\cup j) - v(C) = AI(X_j|X_S; f_C, f_{C \\cup i}).$$\nTherefore, we can leverage \\textit{AI via} to compute the component of the associative importance that enters the model via feature $\\underline{X}_k$. If we were to introduce the feature in isolation, interactions between $\\underline{X}_k$ and other features cannot be restored as they are left perturbed. As we prioritize sensitivity, we instead compare the total importance with the importance, when only the feature of interest is removed. The contribution of the interaction is then attributed to every partaking feature.\n$$\\alpha_{jk}^C = \\alpha_j^C - AI(X_j|X_C \\rightarrow \\underline{X}_{D \\backslash k}; f_C, f_{C \\cup j})$$\nThe SAGE attribution $\\phi_j$ for feature $X_j$ is a linear combination of value function evaluations $\\alpha_j^C$. We yield the component of the SAGE value that can be attributed to feature $\\underline{X}_k$ by replacing the difference terms $\\alpha_j^C$ with the respective $\\alpha_{jk}^C$ values that only account for the contribution of feature $\\underline{X}_k$.\\\\\n\\\\\n\\textbf{Computational complexity:} If the decomposition is to be performed for all $d$ features\/variables, the decomposition scales in $\\mathcal{O}(d^2)$. It is possible to decompose subsets of the features\/variables into the components corresponding to a subset of the features\/variables. I.e., in order to decompose the relevance of one feature\/variable with respect to $m$ components, the number of additional \\textit{DI from}\/\\textit{AI via} evaluations scales in $\\mathcal{O}(m)$. \n\\subsection{DEDACT: Shapley Based Decomposition}\n\\label{subsec:shapley-decomposition}\nIn order to achieve an additive decomposition of associative and direct importance, the game theoretic Shapley values \\cite{shapley1953value} can be adduced. As some recent work elaborates \\cite{Datta2017,lundberg_unified_2017,covert_understanding_2020}, attribution of importance can be seen as a cooperative game in which players (features\/variables) collaborate to yield a certain payoff (importance). As the surplus contribution of a player depends on which players are already in the room, the Shapley value averages the surplus contribution of the feature with respect to all possible configurations. Shapley values yield the unique attribution that satisfies a number of fairness axioms \\cite{shapley1953value} (Appendix \\ref{appendix:shapley}).\\\\\nIn order to apply Shapley values to decompose direct and associative importance measures, we need to define the corresponding value functions $w$. Then, the attribution $\\phi_i$ is defined as\n$$ \\phi_i(w) = \\frac{1}{d} \\sum_{S \\subseteq D \\backslash \\{i\\}} \\binom{d-1}{|S|}^{-1} [w(S \\cup \\{i\\}) - w(S)].$$\n\\textbf{Permutation Feature Importance:} In order to formulate the decomposition of Permutation Feature Importance as a cooperative game, we leverage \\textit{DI from} to quantify the payoff for a team of players $X_J$.\\\\\n$$ w_k(J) := DI(\\underline{X}_k|\\underline{X}_{D \\backslash k} \\leftarrow X_J; f, f)$$\n\\textbf{Conditional SAGE:} The decomposition of the SAGE value function $v$ itself can be formulated as a cooperative game. The corresponding value function $w$ is defined as \n$$ w_j^C(K) := AI(X_j|X_C \\rightarrow \\underline{X}_K; f_C, f_{C \\cup j})$$\n\\textbf{Computational complexity:} As the number of possible subsets grows exponentially in the number of features, the computation of Shapley values is expensive. Like previous work in the field, we suggest to approximate Shapley values by randomly sampling and evaluating orders until the estimates converge \\cite{lundberg_unified_2017,covert_understanding_2020}.\n\n\\section{Simulations}\n\\label{sec:application}\nWe illustrate the usefulness of the approach on two simulated examples. Both provide access to the ground-truth causal graphs, which allows us to validate the interpretation. Furthermore, we chose all relationships to be linear with Gaussian noise, because in this setting reliable conditional distribution estimation is readily available.\\footnote{The code for our experiments is available in an anonoymized repository \\href{https:\/\/github.com\/anonomyzed-submission\/dedact-code}{[click here]}.}\n\\begin{example}[Biomarker Failure]\nWe consider a \\textit{prostate cancer} ($Y$) diagnosis setting. For the model training and evaluation, we don't have access to the true $Y$, but only to labels $L$ that are wrongfully influenced by \\textit{PSA} $P$.\\footnote{The \\textit{prostate specific antigen (PSA)} was wrongfully used in clinical practice for several decades \\cite{ioannidis_biomarker_2013}, leading to over-treatment and its retraction from clinical usage.} In order to avoid bias from \\textit{PSA}, the model is fit with access to two variables only: \\textit{biomarker} $B$ and \\textit{cycling} $C$. However, the model learns to use \\textit{cycling} as a proxy for \\textit{PSA},\\footnote{Studies suggest that cycling increases PSA levels \\cite{jiandani2015effect}.} such that the bias leaks into the model. A fast DEDACT decomposition of the associative influence \\textit{PSA} exposes that the variable leaks into the model via the dependent \\textit{cycling} feature (Figure \\ref{fig:psa-results} (a)). Further, a fast DEDACT decomposition of the PFI of the feature \\textit{cycling} shows that it's importance can be fully attributed to \\textit{PSA}, suggesting the removal of the feature from the model.\n\\label{example:psa-di-vs-ii}\n\\end{example}\n\\begin{figure}[]\n \\centering\n \\input{figures\/figures_decomposition\/wbars_relevance_types}\n\\caption{(a) Associative importance $AI(X_j; f, f)$ and the respective fast decomposition for $PSA$. (b) Direct importance (PFI) and the respective fast decomposition for \\textit{cycling}. (c) The DAG of the ground-truth Structural Causal Model. \\textit{PSA} ($P$) is causal for the historical labels, although only the \\textit{biomarker} ($B$) is relevant for the actual condition. The model uses \\textit{cycling} ($C$) as a proxy, thereby enabling the bias to leak into the model.}\n\\label{fig:psa-results} \n\\end{figure}\n\\begin{example}[Sensitive Attributes]\nIn this hypothetical scenario, adapted from the census income dataset from the UCI Repository \\cite{Dua:2019}, the aim is the to predict the income of subjects. The protected attributes \\textit{age}, \\textit{sex} and \\textit{race} affect the income of subjects directly and indirectly. \\textit{Age} only affects income via \\textit{capital gain}, \\textit{number of educations} and \\textit{hours per week}. \\textit{Race} causes income directly as well as indirectly via \\textit{marriage status} and \\textit{occupation}. And \\textit{sex} influences the target directly as well as indirectly via \\textit{relationship} and \\textit{work class}. The corresponding causal graph is depicted in Figure \\ref{fig:census-income}.\\footnote{We emphasize that this scenario is hypothetical and do not want to suggest that the postulated causal structure reflects the real world.}\\\\\nThe Shapley based DEDACT decomposition of SAGE values for features \\textit{race}, \\textit{sex} and \\textit{age} correctly identifies the features, via which the information from the respective variable enters the model (Figure \\ref{fig:ii-adult-sage}). The Shapley based DEDACT decomposition of PFI correctly identifies the respective sources of prediction-relevant information (Figure \\ref{fig:ii-adult-viafrom}).\n\\end{example}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/figures_decomposition\/ii-adult-sparse_0401_sage_decomp_sage_sub.pdf}\n \\caption{Approximation of the Shapley based decomposition of SAGE values ($60$ SAGE orders, $25$ decomposition orders). The procedure correctly identifies the feature pathways which allow the associative importance to enter the model.}\n \\label{fig:ii-adult-sage}\n\\end{figure}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/figures_decomposition\/ii-adult-sparse_0401_sage_decomp_tdi_decmp_sub.pdf}\n \\caption{Shapley decomposition of PFI ($50$ orders). The sources of prediction relevant information for the features \\textit{education num}, \\textit{workclass} and \\textit{occupation} are correctly identified.}\n \\label{fig:ii-adult-viafrom}\n\\end{figure}\n\\begin{figure}[]\n \\centering\n \\begin{tikzpicture}[thick, scale=1.5, every node\/.style={scale=.7, line width=0.25mm, black, fill=white}]\n \\usetikzlibrary{shapes}\n\t\t\\node[draw, ellipse, scale=0.7] (sex) at (2, .5) {sex};\n\t\t\\node[draw, ellipse, scale=0.7] (age) at (-2, .5) {age};\n\t\t\\node[draw, ellipse, scale=0.7] (race) at (0,.5) {race};\n\t\t\\node[draw, ellipse, scale=0.7] (inc) at (1,-.5) {income};\n\t\n\t \\draw[->,gray] (race) -- (inc);\n \\draw[->,gray] (sex) -- (inc);\n\t\n\t\t\\node[draw, ellipse, scale=0.7] (hpw) at (-1,0) {hours pw};\n\t\t\\node[draw, ellipse, scale=0.7] (ne) at (-2,0) {nr educ};\n\t\t\\node[draw, ellipse, scale=0.7] (wc) at (3,0) {work class};\n\t\t\\node[draw, ellipse, scale=0.7] (oc) at (1,0) {occupation};\n\t\t\\node[draw, ellipse, scale=0.7] (cg) at (-3,0) {capital gain};\n\t\t\\node[draw, ellipse, scale=0.7] (ms) at (0,0) {marriage status};\n\t\t\\node[draw, ellipse, scale=0.7] (rel) at (2,0) {relationship};\n\t\t\n\t\t\\draw[->] (sex) -- (rel);\n\t\t\\draw[->] (sex) -- (wc);\n\t\t\\draw[->] (age) -- (hpw);\n\t\t\\draw[->] (age) -- (cg);\n\t\t\\draw[->] (age) -- (ne);\n\t\t\\draw[->] (race) -- (ms);\n\t\t\\draw[->] (race) -- (oc);\n\n \n \\draw[->, gray] (hpw) -- (inc);\n \\draw[->, gray] (ne) -- (inc);\n \\draw[->, gray] (wc) -- (inc);\n \\draw[->, gray] (oc) -- (inc);\n \\draw[->, gray] (cg) -- (inc);\n \\draw[->, gray] (ms) -- (inc);\n \\draw[->, gray] (rel) -- (inc);\n \n \\end{tikzpicture}\n \\caption{Ground truth DAG for the simulated adult dataset. Gray edges indicate parent edges for predicted income.}\n \\label{fig:census-income}\n\\end{figure}\n\\section{Discussion}\n\nWe propose DEDACT, a general framework that allows to decompose associative importance measures into their direct feature pathway contributions as well as direct importance measures into the contributions of their information sources. Thereby, DEDACT enables novel insight into model and data. DEDACT does not only uncover leakage of information but also explains via which feature pathways the influence enters the model.\\\\\nA fast but non-additive, as well as an additive and fair, but computationally demanding decomposition have been proposed. The approach relies on conditional samples that established methods like conditional SAGE \\cite{covert_understanding_2020} require to compute anyway. Thus, the computational overhead can be reduced.\\\\\nFurther research is required evaluating how well the method scales to more complex distribution and model types. Moreover, the question of how mixed perturbations with different conditioning sets can be coupled to reduce extrapolation is a promising avenue for future research. We conjecture that methods on information preserving independent representation learning could be leveraged. In addition, future work will assess how knowledge about the dependence structure in the graph can be used to reduce the computational complexity of the method. We see DEDACT as a step towards understanding machine learning methods within the context they operate in: the underlying data generating mechanism.\\\\\n\n \\bibliographystyle{splncs04}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWith growing diversity in personal food preference and regional cuisine style, personalized information systems that can transform a recipe into any selected regional cuisine style that a user might prefer would help food companies and professional chefs create new recipes.\n\n\nTo achieve this goal, there are two significant challenges: 1) identifying the degree of regional cuisine style mixture of any selected recipe; and 2) developing an algorithm that shifts a recipe into any selected regional cuisine style. \n\nAs to the former challenge, with growing globalization and economic development, it is becoming difficult to identify a recipe's regional cuisine style with specific traditional styles since regional cuisine patterns have been changing and converging in many countries throughout Asia, Europe, and elsewhere \\cite{khoury2014increasing}. Regarding the latter challenge, to the best of our knowledge, little attention has been paid to developing algorithms which transform a recipe's regional cuisine style into any selected regional cuisine pattern, cf. \\cite{pinel2014using,pinel2014substitution}. Previous studies have focused on developing an algorithm which suggests replaceable ingredients based on cooking action \\cite{shidochi2009finding}, degree of similarity among ingredient \\cite{nozawa}, ingredient network \\cite{teng2012recipe}, degree of typicality of ingredient \\cite{yokoi2015typicality}, and flavor (foodpairing.com).\n\nThe aim of this study is to propose a novel data-driven system for transformation of regional cuisine style. This system has two characteristics. First, we propose a new method for identifying a recipe's regional cuisine style mixture by calculating the contribution of each ingredient to certain regional cuisine patterns, such as Mediterranean, French, or Japanese, by drawing on ingredient prevalence data from large recipe repositories. Also the system visualizes a recipe's regional cuisine style mixture in two-dimensional space under barycentric coordinates using what we call a {\\it Newton diagram}. Second, the system transforms a recipe's regional cuisine pattern into any selected regional style by recommending replaceable ingredients in existing recipes.\n\nAs an example of this proposed system, we transform a traditional Japanese recipe, Sukiyaki, into French style.\n\n\\section{Architecture of transformation system}\nFigure \\ref{fig1} shows the overall architecture of the transformation system, which consists of two steps: 1) identification and visualization of a recipe's regional cuisine style mixture; and 2) algorithm which transforms a given recipe into any selected regional\/country style. Details of the steps are described as follows.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{fig1}\n\\end{center}\n\\caption{ Architecture of transformation system which transform a given recipe into any selected country\/region style}\\label{fig1}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=15cm]{fig2}\n\\end{center}\n\\caption{ Neural network model for predicting regional cuisine from list of ingredients.}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Step 1: Identification and visualization of a recipe's regional cuisine style mixture}\nUsing a neural network method as detailed below, we identify a recipe's regional cuisine style. The neural network model was constructed as shown in Figure \\ref{fig2}. The number of layers and dimension of each layer are also shown in Figure \\ref{fig2}. \n\nWhen we enter a recipe, this model classifies which country or regional cuisine the recipe belongs to. The input is a vector with the dimension of the total number of ingredients included in the dataset, and only the indices of ingredients contained in the input recipe are 1, otherwise they are 0. \n\nThere are two hidden layers. Therefore, this model can consider a combination of ingredients to predict the country probability. Dropout is also used for the hidden layer, randomly (20\\%) setting the value of the node to 0. So a robust network is constructed. The final layer's dimension is the number of countries, here 20 countries. In the final layer, we convert it to a probability value using the softmax function, which represents the probability that the recipe belongs to that country. \nADAM \\cite{kingma2014adam} was used as an optimization technique. The number of epochs in training was 200. These network structure and parameters were chosen after preliminary experiments so that the neural network could perform the country classification task as efficiently as possible.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Statistics of Yummly dataset and some recipe examples.}\n\\begin{tabular}{cc}\n\\begin{minipage}{0.5\\hsize}\n\\begin{center}\n\\begin{tabular}{|l|c|r|}\n\\hline\nCountry &Recipes & Ingredients \\\\ \\hline \\hline\nItalian\t&\t7838\t&\t2929\t\\\\ \\hline\nMexican\t&\t6438\t&\t2684\t\\\\ \\hline\nSouthern US\t&\t4320\t&\t2462\t\\\\ \\hline\nIndian\t&\t3003\t&\t1664\t\\\\ \\hline\nChinese\t&\t2673\t&\t1792\t\\\\ \\hline\nFrench\t&\t2646\t&\t2102\t\\\\ \\hline\nCajun Creole\t&\t1546\t&\t1576\t\\\\ \\hline\nThai\t&\t1539\t&\t1376\t\\\\ \\hline\nJapanese\t&\t1423\t&\t1439\t\\\\ \\hline\nGreek\t&\t1175\t&\t1198\t\\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\n\\begin{minipage}{0.5\\hsize}\n\\begin{tabular}{|l|c|r|}\n\\hline\nCountry &Recipes & Ingredients \\\\ \\hline \\hline\nSpanish\t&\t989\t&\t1263\t\\\\ \\hline\nKorean\t&\t830\t&\t898\t\\\\ \\hline\nVietnamese\t&\t825\t&\t1108\t\\\\ \\hline\nMoroccan\t&\t821\t&\t974\t\\\\ \\hline\nBritish\t&\t804\t&\t1166\t\\\\ \\hline\nFilipino\t&\t755\t&\t947\t\\\\ \\hline\nIrish\t&\t667\t&\t999\t\\\\ \\hline\nJamaican\t&\t526\t&\t877\t\\\\ \\hline\nRussian\t&\t489\t&\t872\t\\\\ \\hline\nBrazilian\t&\t467\t&\t853\t\\\\ \\hline \\hline\nALL\t&\t39774\t&\t6714\t\\\\ \n\\hline\n\\end{tabular}\n\\end{minipage}\n\n\\end{tabular}\n\n\\begin{minipage}{0.9\\hsize}\n\\begin{tabular}{|l|c|l|}\n\\hline\nRecipeID &Country& Ingredients \\\\ \\hline \\hline\n34466 & British & greek yogurt, lemon curd, confectioners sugar, raspberries\\\\ \\hline\n44500 & Indian & chili, mayonaise, chopped onion, cider vinegar, fresh mint, cilantro leaves\\\\ \\hline\n38233 & Thai & sugar, chicken thighs, cooking oil, fish sauce, garlic, black pepper\\\\ \\hline\n\\end{tabular}\n\\end{minipage}\n\n\\label{table1}\n\\end{center}\n\\end{table}\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Example of ingredient classification by the neural network. Three top countries are listed with the probability that the ingredient are classified into.}\n\\begin{tabular}{|c|c|r|r|r|}\n\\hline\n Ingredient & Top1 & Top2 & Top3\\\\ \\hline \\hline\nOnions & French & Italian & Mexican \\\\ \n & 0.130 & 0.126 & 0.126 \\\\ \\hline\nSoy sauce & Japanese & Chinese & Filipino \\\\ \n & 0.246 & 0.233 & 0.122 \\\\ \\hline\n Mirin & Japanese & French & Korean \\\\ \n & 0.890 & 0.040 &0.020 \\\\ \n\\hline\n\\end{tabular}\n\\label{table1a}\n\\end{center}\n\\end{table}\n\n\nIn this study, we used a labeled corpus of Yummly recipes to train this neural network. Yummly dataset has 39774 recipes from the 20 countries as shown in Table \\ref{table1}. Each recipe has the ingredients and country information. Firstly, we randomly divided the data set into 80\\% for training the neural network and 20\\% for testing how precisely it can classify. The final neural network achieved a classification accuracy of 79\\% on the test set. Figure \\ref{fig.conf} shows the confusion matrix of the neural network classifficaiton. Table \\ref{table1a} shows the examples of ingredient classification results. Common ingredients, onions for example, that appear in many regional recipes are assigned to all countries with low probability. On the other hands some ingredients that appear only in specific country are assigned to the country with high probability. For example mirin that is a seasoning commonly used in Japan is classified into Japan with high probability.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{confusion\n\\end{center}\n\\caption{Confusion matrix of neural network classiffication. }\\label{fig.conf}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{fig3\n\\end{center}\n\\caption{ Newton diagram: visualization of probability that the recipe belongs to the several regional cuisine style. Countries are placed by spectral drawing.\n}\\label{fig3}\n\\end{figure}\n\n\nBy using the probability values that emerge from the activation function in the neural network, rather than just the final classification, we can draw a barycentric Newton diagram, as shown in Figure \\ref{fig3}. The basic idea of the visualization, drawing on Isaac Newton's visualization of the color spectrum \\cite{newton1704}, is to express a mixture in terms of its constituents as represented in barycentric coordinates. This visualization allows an intuitive interpretation of which country a recipe belongs to. If the probability of Japanese is high, the recipe is mapped near the Japanese. The countries on the Newton diagram are placed by spectral graph drawing \\cite{koren2003spectral}, so that similar countries are placed nearby on the circle. The calculation is as follows.\nFirst we define the adjacency matrix $W$ as the similarity between two countries. \nThe similarity between country $i$ and $j$ is calculated by cosine similarity of county $i$ vector and $j$ vector. These vector are defined in next section. $W_{ij} = sim(vec_i, vec_j)$.\nThe degree matrix $D$ is a diagonal matrix where $D_{ii} = \\sum_{j} W_{ij}$. \nNext we calculate the eigendecomposition of $D^{-1}W$. The second and third smallest eingenvalues and corresponded eingevectors are used for placing the countries. Eigenvectors are normalized so as to place the countries on the circle. \n\n\\subsection{Step 2: Transformation algorithm for transforming regional cuisine style}\nIf you want to change a given recipe into a recipe having high probability of a specific country by just changing one ingredient, which ingredient should be alternatively used?\n\nWhen we change the one ingredient $x_i$ in the recipe to ingredient $x_j$, the probability value of country likelihood can be calculated by using the above neural network model. If we want to change the recipe to have high probability of a specific country $c$, we can find ingredient $x_j$ that maximizes the following probability. $P(C=c|r - x_i + x_j)$\nwhere $r$ is the recipe.\nHowever, with this method, regardless of the ingredient $x_i$, only specific ingredients having a high probability of country $c$ are always selected. In this system, we want to select ingredients that are similar to ingredient $x_i$ and have a high probability of country $c$. Therefore, we propose a method of extending word2vec as a method of finding ingredients resembling ingredient $x_i$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{word2vec_arch\n\\end{center}\n\\caption{The word2vec (skip-gram) architecture. The left panel is the traditional word2vec with window size $n=2$. The middle panel is the word2vec for recipe data. The right panel is the word2vec for recipe data with country information. \n}\\label{word2vec_arch}\n\\end{figure}\n\n\nWord2vec is a technique proposed in the field of natural language processing \\cite{mikolov2013distributed}. As the name implies, it is a method to vectorize words, and similar words are represented by similar vectors. To train word2vec, skip-gram model is used. In the skip-gram model, the objective is to learn word vector representations that can predict the nearby words. The objective function is \n\n\\begin{equation}\n\\sum_{d \\in D} \\sum_{w_i \\in d} \\sum_{-n \\leq j \\leq n, j \\neq 0} \\log P(w_{i + j}|w_i) \\label{eq:02}\n\\end{equation}\nwhere $D$ is the set of documents, $d$ is a document, $w_i$ is a word, and $n$ is the window size. This model predicts the $n$ words before and after the input word, as described in left side of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the surrounding word $w_{i+j}$ given the center word $w_i$.\nThe probability is \n\\begin{equation}\nP(w_j|w_i) = \\frac{\\exp(v_{w_i}^Tv_{w_j}^{'})}{\\sum_{w \\in W} \\exp(v_{w_i}^Tv_w^{'})}\n\\end{equation}\nwhere $v_w \\in \\mathbb{R}^K$ is an input vector of word $w$, $v^{'}_w \\in \\mathbb{R}^K$ is an output vector of word $w$, $K$ is the dimension of the vector, and $W$ is the set of all words. To optimize this objective function, hierarchical softmax or negative sampling method \\cite{mikolov2013distributed} are used. After that we get the vectors of words and we can calculate analogies by using the vectors. For example, the analogy of ``King - Man + Women = ?\" yields ``Queen\" by using word2vec.\n\nIn this study, word2vec is applied to the data set of recipes. Word2vec can be applied by considering recipes as documents and ingredients as words. We do not include a window size parameter, since it is used to encode the ordering of words in document where it is relevant. In recipes, the listing of ingredients is unordered. The objective function is \n\\begin{equation}\n\\sum_{r \\in R} \\sum_{w_i \\in r} \\sum_{j \\neq i} \\log P(w_{j}|w_i) \\label{eq:02}\n\\end{equation}\nwhere $R$ is a set of recipes, $r$ is a recipe, and $w_i$ is the $i$th ingredient in recipe $r$. The architecture is described in middle of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the ingredient $w_j$ in the same recipe given the ingredient $w_i$. The probability is defined below.\n\\begin{equation}\nP(w_j|w_i) = \\frac{\\exp(v_{w_i}^Tv_{w_j}^{'})}{\\sum_{w \\in W} \\exp(v_{w_i}^Tv_w^{'})}\n\\end{equation}\nwhere $w$ is an ingredient, $v_w \\in \\mathbb{R}^K$ is an input vector of ingredient, $v^{'}_w \\in \\mathbb{R}^K$ is an output vector of ingredient, $K$ is the dimension of the vector, and $W$ is the set of all ingredients.\n\nEach ingredient is vectorized by word2vec, and the similarity of each ingredient is calculated using cosine similarity. Through vectorization in word2vec, those of the same genre are placed nearby. In other words, by using the word2vec vector, it is possible to select ingredients with similar genres. \n\nNext, we extend word2vec to be able to incorporate information of the country. When we vectorize the countries, we can calculate the analogy between countries and ingredients. For example, this method can tell us what is the French ingredient that corresponds to Japanese soy sauce by calculating ``Soy sauce - Japan + French = ?\".\n\nThe detail of our method is as follows. We maximize objective function (\\ref{eq:02}). \n\n\\begin{equation}\n\\sum_{r \\in R} \\sum_{w_i \\in r} \\left( \\log P(w_{i}|c_r) + \\log P(c_r|w_{i}) + \\sum_{j \\neq i} \\log P(w_{j}|w_i)\\right)\\label{eq:02}\n\\end{equation}\nwhere $R$ is a set of recipes, $r$ is a recipe, $w_i$ is the $i$th ingredient in recipe $r$, and $c_r$ is the country recipe $r$ belongs to. The architecture is described in right of Figure \\ref{word2vec_arch}. The objective function is to maximize the likelihood of the prediction of the ingredient $w_j$ in the same recipe given the ingredient $w_i$ along with the prediction of the the ingredients $w_i$ given the country $c_r$ and the prediction of the the country $c_r$ given the ingredient $w_i$. The probability is defined below.\n\n\\begin{equation}\nP(b|a) = \\frac{\\exp(v_{a}^Tv_{b}^{'})}{\\sum_{c \\in W} \\exp(v_{a}^Tv_c^{'})}\n\\end{equation}\nwhere $a$ is a ingredient or country, $b,c$ are also, $v_a \\in \\mathbb{R}^K$ is an input vector of ingredient or country, $v^{'}_a \\in \\mathbb{R}^K$ is an output vector of ingredient or country, $K$ is the dimension of vector, and $W$ is the set of all ingredients and all countries.\n\nWe can use hierarchical softmax or negative sampling \\cite{mikolov2013distributed} to maximize objective function (\\ref{eq:02}) and find the vectors of ingredients and countries in the same vector space. \n\nTable \\ref{table2} shows the ingredients around each country in the vector space, and which could be considered as most authentic for that regional cuisine \\cite{ahn2011flavor}. Also, Figure \\ref{fig4} shows the ingredients and countries in 2D map by using t-SNE method \\cite{maaten2008visualizing}.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Authentic ingredients for each country. Top 5 ingredients around each country in the vector space.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n & French & Japanese & Italian & Mexican \\\\ \\hline \\hline\nTop1 & Cognac & Mirin & Grated parmesan cheese & Corn tortillas\\\\ \\hline\nTop2 & Calvados & Dashi & pecorino romano cheese & Salsa\\\\ \\hline\nTop3 & Thyme springs & Nori & prosciutto & Tortilla chips\\\\ \\hline\nTop4 & Gruyere cheese & Wasabi paste &marinara sauce & Guacamole\\\\ \\hline\nTop5 & Nicoise olives & Bonito flakes &Sweet italian sausage& Poblano peppers \\\\ \n\\hline\n\\end{tabular}\n\\label{table2}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=15cm]{word2vec\n\\end{center}\n\\caption{Ingredients and countries map by extended word2vec: Each ingredient and country is mapped in 2D by using t-SNE. Also Each ingredient is colored by using t-SNE to convert 100 dimension vector into 3 dimension. The 3 dimension is corresponded to RGB color. Countries are represented by bold black.}\\label{fig4}\n\\end{figure}\n\n\n\n\\section{Experiment}\nOur substitution strategy is as follows.\nFirst we use extended word2vec and train it by Yummly dataset. After that all ingredients and countries are vectorized into 100 dimensional vector space. Second we find substitution by analogy calculation. For example, to find french substitution of Mirin, we calculate ``Mirin - Japanese + French\" in the vector space and get the vector as result. After that we find similar ingredients around the vector by calculating the cosine similarity.\n\nAs an example of our proposed system, we transformed a traditional Japanese ``Sukiyaki\" into French style. Table \\ref{table3} shows the suggested replaceable ingredients and the probability after replacing. ``Sukiyaki\" consists of soy sauce, beef sirloin, white sugar, green onions, mirin, shiitake, egg, vegetable oil, konnyaku, and chinese cabbage. Figure \\ref{fig:sukiyaki_french} shows the Sukiyaki in French style cooked by professional chef KM who is one of the authors of this paper. He assesses the new recipe as valid and novel to him in terms of Sukiyaki in French. \nHere our task is in generating a new dish, for which by definition there is no ground truth for comparison.\nRating by experts is the standard approach for assessing novel generative artifacts, e.g. in studies of creativity \\cite{jordanous2012standardised}, but going forward it is important to develop other approaches for assessment.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Alternative ingredients suggested by extended word2vec model and country probability of changing food ingredients in order from the top. Professional chef KM who is one of the authors of this paper chose one alternative ingredient from top 10 suggested ingredients each.}\n\\begin{tabular}{|c|c|r|r|r|}\n\\hline\n Original Ingredient & Alternative Ingredient & P(Japanese) & P(French)&\\# of replacement \\\\ \\hline \\hline\n- & - & 0.974 & 0.000 & 0\\\\ \\hline\nMirin & Calvados & 0.552 & 0.009 & 1\\\\ \\hline\nVegetable oil & Olive oil & 0.393 & 0.031 & 2\\\\ \\hline\nSoy sauce & Bouquet garni & 0.011 & 0.976 & 3\\\\ \\hline\nGreen onions& Fresh tarragon & 0.000 &0.997 & 4 \\\\ \\hline\nEgg & Melted butter& 0.000 &0.999 & 5 \\\\ \n\\hline\n\\end{tabular}\n\\label{table3}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french0\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french1\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french2\n\\end{minipage}%\n\\begin{minipage}[b]{.25\\linewidth}\n\\centering\\includegraphics[width=3.4cm, angle=90]{sukiyaki_french3\n\\end{minipage}%\n\\caption{Sukiyaki in French style. Professional chef KM who is one of the authors of this paper cooked the recipe suggested by our system.}\\label{fig:sukiyaki_french}\n\\end{figure}\n\n\n\\section{Discussion}\nWith growing diversity in personal food preference and regional cuisine style, the development of data-driven systems which can transform recipes into any given regional cuisine style might be of value for food companies or professional chefs to create new recipes.\n\nIn this regard, this study adds two important contributions to the literature. First, this is to the best of our knowledge, the first study to identify a recipe's mixture of regional cuisine style from the large number of recipes around the world. Previous studies have focused on assessing degree of adherence to a single regional cuisine pattern. For example, Mediterranean Diet Score is one of the most popular diet scores. This method uses 11 main items (e.g., fruit, vegetable, olive oil, and wine) as criteria for assessing the degree of one's Mediterranean style \\cite{panagiotakos2006dietary}. However, in this era, it is becoming difficult to identify a recipe's regional cuisine style with specific country\/regional style. \nFor example, should Fish Provencal, whose recipe name is suggestive of Southern France, be cast as French style? The answer is mixture of different country styles: 32\\% French; 26\\% Italian; and 38\\% Spanish (see Figure \\ref{fig3}).\n\nFurthermore, our identification algorithm can be used to assess the degree of personal regional cuisine style mixture, using the user's daily eating pattern as inputs. For example, when one enters the recipes that one has eaten in the past week into the algorithm, the probability values of each country would be returned, which shows the mixture of regional cuisine style of one's daily eating pattern. As such, a future research direction would be developing algorithms that can transform personal regional cuisine patterns to a healthier style by providing a series of recipes that are in accordance with one's unique food preferences.\n\nOur transformation algorithm can be improved by adding multiple datasets from around the world. Needless to say, lack of a comprehensive data sets makes it difficult to develop algorithms for transforming regional cuisine style. For example, Yummly, one of the largest recipe sites in the world, is less likely to contain recipes from non-Western regions. Furthermore, data on traditional regional cuisine patterns is usually described in its native language. As such, developing a way to integrate multiple data sets in multiple languages is required for future research. \n\nOne of the methods to address this issue might be as follows: 1) generating the vector representation for each ingredient by using each data set independently; 2) translating only a small set of common ingredients among each data set, such as potato, tomato, and onion; 3) with a use of common ingredients, mapping each vector representation into one common vector space using a canonical correlation analysis \\cite{kettenring1971canonical}, for example. \n\nSeveral fundamental limitations of the present study warrant mention. First of all, our identification and transformation algorithms depend on the quantity and quality of recipes included in the data. As such, future research using our proposed system should employ quality big recipe data.\nSecond, the evolution of regional cuisines prevents us from developing precise algorithm. For example, the definition of Mediterranean regional cuisine pattern has been revised to adapt to current dietary patterns \\cite{serra2004does,kinouchi2008non}. Therefore, future research should employ time-trend recipe data to distinctively specify a recipe's mixture of regional cuisine style and its date cf.~\\cite{varshney2013flavor}. \nThird, we did not consider the cooking method (e.g., baking, boiling, and deep flying) as a characteristic of country\/regional style. Each country\/region has different ways of cooking ingredients and this is one of the important factors characterizing the food culture of each country\/region. \nFourth, the combination of ingredients was not considered as the way to represent country\/regional style. For example, previous studies have shown that Western recipes and East Asian recipes are opposite in flavor compounds included in the ingredient pair \\cite{zhu2013geography,varshney2013flavor,jain2015spices,tallab2016exploring,ahn2011flavor}. For example, Western cuisines tend to use ingredient pairs sharing many flavor compounds, while East Asian cuisines tend to avoid compound sharing ingredients. It is suggested that combination of flavor compounds was also elemental factor to characterize the food in each country\/region. As such, if we analyze the recipes data using flavor compounds, we might get different results.\n\n\n\nIn conclusion, we proposed a novel system which can transform a given recipe into any selected regional cuisine style. This system has two characteristics: 1) the system can identify a degree of regional cuisine style mixture of any selected recipe and visualize such regional cuisine style mixture using a barycentric Newton diagram; 2) the system can suggest ingredient substitution through extended word2vec model, such that a recipe becomes more authentic for any selected regional cuisine style. Future research directions were also discussed.\n\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that they have no conflict of interest.\n\n\\section*{Author Contributions}\nMK, LRV, and YI had the idea for the study and drafted the manuscript. MK performed the data collection and analysis. MS, CH, and KM participated in the interpretation of the results and discussions for manuscript writing and finalization. All authors read and approved the final manuscript.\n\n\n\\section*{Funding}\nVarshney's work was supported in part by the IBM-Illinois Center for Cognitive\nComputing Systems Research (C3SR), a research collaboration as part of the\nIBM AI Horizons Network.\n\n\n\\section*{Acknowledgments}\nThis study used data from Yummly. We would like to express our deepest gratitude to everyone who participated in this services. We thank Kush Varshney for suggesting the spectral graph drawing approach to placing countries on the circle.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}