0$ for $|x|>0$, $\varphi''(0)>0$ and $\varphi(x)\,\, {\vtop{\ialign{#\crcr\rightarrowfill\crcr \noalign{\kern0pt\nointerlineskip}\hglue3.pt${\scriptstyle {x\to\infty}}$\hglue3.pt\crcr}}}\,\,+\infty$ (in other words a $1$--dimensional system in a confining potential). There is only one motion per energy value (up to a shift of the initial datum along its trajectory) and all motions are periodic so that the system is {\it monocyclic}. Assume also that the potential $\varphi(x)$ depends on a parameter $V$. One defines {\it state} a motion with given energy $E$ and given $V$. And: \* \halign{#\ $=$\ & #\hfill\cr $U$ & total energy of the system $\equiv K+\varphi$\cr $T$ & time average of the kinetic energy $K$\cr $V$ & the parameter on which $\varphi$ is suposed to depend\cr $p$ & $-$ time average of $\partial_V \varphi$\cr} \* \noindent{}A state is parameterized by $U,V$ and if such parameters change by $dU, dV$ respectively we define: $$dL=-p dV,\qquad dQ=dU+p dV\Eqa(A2.1)$$ % then: \* \noindent{}{\sl Theorem} (Helmholtz): {\it the differential $({dU+pdV})/{T}$ is exact.} \* In fact let: $$S=2\log \int_{x_-(U,V)}^{x_+(U,V)}\sqrt{K(x;U,V)}dx= 2\log \int_{x_-(U,V)}^{x_+(U,V)}\sqrt{U-\varphi(x)}dx\Eqa(A2.2)$$ % ($\fra12S$ is the logarithm of the action), so that: $$S=\fra{\int (dU-\partial_V\varphi(x) dV) \fra{dx}{\sqrt{K}}}{ \int K\fra{dx}{\sqrt{K}}}\Eqa(A2.3)$$ % and, noting that $\fra{dx}{\sqrt K} =\sqrt{\fra2m} dt$, we see that the time averages are given by integrating with respect to $\fra{dx}{\sqrt K}$ and dividing by the integral of $\fra{1}{\sqrt K}$. We find therefore: $$dS=\fra{dU+p dV}{T}\Eqa(A2.4)$$ % Boltzmann saw that this was not a simple coincidence: his interesting (and healthy) view of the continuum (which he probably never really considered more than a convenient artifact, useful for computing quantities describing a discrete world) led him to think that in some sense {\it monocyclicity was not a strong assumption}. In general one can call {\it monocyclic} a system with the property that there is a curve $\ell\to x(\ell)$, parameterized by its curvilinear abscissa $\ell$, varying in an interval $0< \ell< L(E)$, closed and such that $ x(\ell)$ covers all the positions compatible with the given energy $E$. Let $ x= x(\ell)$ be the parametric equations so that the energy conservation can be written: $$\fra12 m\dot\ell^2+\varphi(x(\ell))=E \Eqa(A2.5)$$ % then if we suppose that the potential energy $\varphi$ depends on a parameter $V$ and if $T$ is the average kinetic energy, $p=-\langle{\partial_V \varphi}\rangle$ it is, for some $S$: % $$dS=\fra{dE+pdV}{T},\qquad p=-\langle{\partial_V \varphi}\rangle,\quad T=\langle{K}\rangle\Eqa(A2.6)$$ % where $\langle \cdot\rangle$ denotes time average. The above can be applied to a gas in a box. Imagine the box containing the gas to be covered by a piston of section $A$ and located to the right of the origin at distance $L$: so that $V=AL$. The microscopic model for the pistion will be a potential $\lis\f(L-\x)$ if $x=(\x,\h,\z)$ are the coordinates of a particle. The function $\lis\f(r)$ will vanish for $r>r_0$, for some $r_0$, and diverge to $+\io$ at $r=0$. Thus $r_0$ is the width of the layer near the piston where the force of the wall is felt by the particles that happen to roam there. Noting that the potential energy due to the walls is $\f=\sum_j \lis\f(L-\x_j)$ and that $\dpr_V \f=A^{-1}\dpr_L\f$ we must evaluate the time average of: $$\dpr_L \f(x)=-\sum_j \lis\f'(L-\x_j)\Eqa(A2.7)$$ % As time evolves the particles with $\x_j$ in the layer within $r_0$ of the wall will feel the force exercized by the wall and bounce back. Fixing the attention on one particle in the layer we see that it will contribute to the average of $\dpr_L \f(x)$ the amount: $$\fra1{\rm total\ time} 2\ig_{t_0}^{t_1}- \lis\f'(L-\x_j) dt\Eqa(A2.8)$$ % if $t_0$ is the first instant when the point $j$ enters the layer and $t_1$ is the instante when the $\x$--compoent of the velocity vanishes ``against the wall''. Since $-\lis\f'(L-\x_j)$ is the $\x$--component of the force, the integral is $-2m|\dot\x_j|$ (by Newton's law), provided $\dot\x_j>0$ of course. The number of such contributions to the average per unit time are therefore given by $\r_{wall}\, A\, \ig_{v>0} 2mv\, f(v)\, v\, dv$ if $\r_{wall}$ is the density (average) of the gas near the wall and $f(v)$ is the fraction of particles with velocity between $v$ and $v+dv$. Using the ergodic hypothesis (\ie the microcanonical ensemble) and the equivalence of the ensembles to evaluate $f(v)$ it follows that: $$p\defi \media{\dpr_V\f}= \r_{wall} \b^{-1}\Eqa(A2.9)$$ % where $\b^{-1}=k_B T$ with $T$ the absolute temperature and $k_B$ the Boltmann's constant. That the \equ(A2.9) yields the correct value of the pressure is well known, see [MP], in Classical Statistical Mechanics; in fact often it is even taken as microscopic definition of the pressure. \* \0{\bf Appendix A3. A proof of the fluctuation theorem.} \numsec=3\numfor=1\* \0{\it(A) Description of the SRB statistics.} \* A set $E$ is a {\it rectangle} with {\it center} $x$ and {\it axes} $\D^u,\D^s$ if:\\ 1) $\D^u,\D^s$ are two connected surface elements of $W^u_x,W^s_x$ containing $x$.\\ 2) for any choice of $\x\in\D^u$ and $\h\in\D^s$ the local manifolds $W^{s,\d}_\x$ and $W^{u,\d}_\h$ intersect in one and only one point $x(\x,\h)=W^{s,\d}_\x\cap W^{u,\d}_\h$. The point $x(\x,\h)$ will also be denoted $\x\times\h$.\\ 3) the boundaries $\dpr\D^u$ and $\dpr\D^s$ (regarding the latter sets as subsets of $W^u_x$ and $W^s_x$) have zero surface area on $W^u_x$ and $W^s_x$.\\ 4) $E$ is the set of points $\D^u\times\D^s$. Note that {\it any} $x'\in E$ can be regarded as the center of $E$ because there are $\D^{\prime u},\D^{\prime s}$ both containing $x'$ and such that $\D^u\times\D^s\= \D^{\prime u}\times\D^{\prime s}$. Hence each $E$ can be regarded as a rectangle centered at any $x'\in E$ (with suitable axes). See figure. \figini{figurediffxxx.ps} \8/punto { gsave \83 0 360 newpath arc fill stroke grestore} def \8/puntino { gsave \82 0 360 newpath arc fill stroke grestore} def \8 \8/origine1assexper2pilacon|P_2-P_1| { \84 2 roll 2 copy translate exch 4 1 roll sub \83 1 roll exch sub 2 copy atan rotate 2 copy \8exch 4 1 roll mul 3 1 roll mul add sqrt } def \8 \8/punta0{0 0 moveto dup dup 0 exch 2 div lineto 0 \8lineto 0 exch 2 div neg lineto 0 0 lineto fill \8stroke } def \8 \8/dirpunta{ \8gsave origine1assexper2pilacon|P_2-P_1| \8 0 translate 7 punta0 grestore} def \8 \8/uno{% a v \8dup mul div neg} def \8/due{% T v b \8sub div} def \8/p{% a b T v \8dup 5 1 roll 3 -1 roll due 3 1 roll exch uno add} def \8 \8gsave \840 40 40 0 360 arc \820 40 moveto 60 40 lineto \840 20 moveto 40 60 lineto \8stroke \8 \8140 40 40 0 360 arc \8120 40 moveto 160 40 lineto \8140 20 moveto 140 60 lineto \8stroke \8150 40 2 0 360 arc fill \8140 50 2 0 360 arc fill \8150 50 2 0 360 arc fill \8stroke \8175 75 moveto 160 60 lineto \8175 75 160 60 dirpunta \8150 10 moveto 150 70 lineto \8110 50 moveto 165 50 lineto \8stroke \8 \8240 40 40 0 360 arc \8220 20 moveto 220 60 lineto \8260 60 lineto 260 20 lineto 260 20 220 20 lineto \8220 40 moveto 260 40 lineto \8240 20 moveto 240 60 lineto \8stroke \8220 40 puntino \8260 40 puntino \8240 60 puntino \8240 20 puntino \8grestore \figfin \eqfig{260}{90}{ \ins{43}{37}{$x$} \ins{43}{60}{$\D^s$} \ins{60}{40}{$\D^u$} \ins{155}{36}{$\x$} \ins{130}{60}{$\h$} \ins{177}{77}{$\x\times\h$} \ins{245}{70}{$E$} }{figurediffxxx.ps}{} \* {\nota \0The circle is a small neighborhood of $x$; the first picture shows the axes; the intermediate picture shows the $\times$ operation and $W^{u,\d}_\h, W^{s,\d}_\x$; the third picture shows the rectangle $E$ with the axes and the four marked points are the boundaries $\dpr\D^u$ and $\dpr\D^s$. The picture refers to a two dimensional case and the stable and unstable manifolds are drawn as flat, \ie the $\D$'s are very small compared to the curvature of the manifolds. The center $x$ is drawn in a central position, but it can be {\it any} other point of $E$ provided $\D^u$ and $\D^s$ are correspondingly redefined. One should meditate on the symbolic nature of the drawing in the cases of higher dimension.\vfill} The {\it unstable boundary} of a rectangle $E$ will be the set $\dpr_u E=\D^u\times\dpr\D^s$; the {\it stable boundary} will be $\dpr_s E=\dpr\D^u\times\D^s$. The boundary $\dpr E$ is therefore $\dpr E=\dpr_s E\cup\dpr_u E$. The set of the {\it interior points} of $E$ will be denoted $E^0$. A {\it pavement} of $ M$ will be a covering $\EE=(E_1,\ldots,E_\NN)$ of $ M$ by $\NN$ rectangles with pairwise disjoint interiors. The {\it stable (or unstable) boundary} $\dpr_s\EE$ (or $\dpr_u \EE$) of $\EE$ is the union of the stable (or unstable) boundaries of the rectangles $E_j$: $\dpr_u \EE=\cup_j\dpr_u E_j$ and $\dpr_s \EE=\cup_j\dpr_s E_j$. A pavement $\EE$ is called {\it markovian} if its stable boundary $\dpr_s \EE$ retracts on itself under the action of $S$ and its unstable boundary retracts on itself under the action of $S^{-1}$, [Si], [Bo], [Ru1]; this means: $$S\dpr_s\EE\subseteq \dpr_s\EE,\qquad S^{-1}\dpr_u\EE\subseteq \dpr_u\EE \Eqa(A3.1)$$ \0Setting $M_{j,j'}=0$, $j,j'\in\{1,\ldots,\NN\}$, if $S E^0_j\cap E^0_{j'}=\emptyset$ and $M_{j,j'}=1$ otherwise we call $C$ the set of sequences $\V j=(j_k)_{k=-\io}^\io$, $j_k\in\{1,\ldots,\NN\}$ such that $M_{j_k,j_{k+1}}\=1$. The transitivity of the system $( M,S)$ implies that $M$ is {\it transitive}: \ie there is a power of the matrix $M$ with all entries positive. The space $C$ will be called the space of the {\it compatible symbolic sequences}. If $\EE$ is a markovian pavement and $\d$ is small enough the map: $$X: \V j\in C\,\to\,x=\bigcap_{k=-\io}^\io S^{-k} E_{j_k}\in M\Eqa(A3.2)$$ \0is continuous and $1-1$ between the complement $ M_0\subset M$ of the set $N= \cup_{k=-\io}^\io S^k\dpr \EE $ and the complement $C_0\subset C$ of $X^{-1}(N)$. This map is called the {\it symbolic code} of the points of $ M$: it is a code that associates with each $x\not\in N$ a sequence of symbols $\V j$ which are the labels of the rectangles of the pavement that are successively visited by the motion $S^jx$. The symbolic code $X$ transforms the action of $S$ into the {\it left shift} $\th$ on $C$: $S X(\V j)= X(\th \V j)$. A key result, [Si], is that it transforms the {\it volume measure} $\m_0$ on $ M$ into a {\it Gibbs distribution}, [LR], [Ru2], $\lis\m_0$ on $C$ with formal Hamiltonian: $$H(\V j)=\sum_{k=-\io}^{-1} h_-(\th^k\V j)+h_0(\V j)+\sum_{k=0}^\io h_+(\th^k \V j)\Eqa(A3.3)$$ \0where, see \equ(2.1): $$\eqalign{ h_-(\V j)=&-\log \L_s(X(\V j)),\quad h_+(\V j)=\log \L_u(X(\V j)),\cr h_0(\V j)=&-\log\sin\a(X(\V j))\cr}\Eqa(A3.4)$$ If $F$ is H\"older continuous on $ M$ the function $\lis F(\V j)=F(X(\V j))$ can be represented in terms of suitable functions $\F_k(j_{-k},\ldots,j_k)$ as: $$\lis F(\V j)=\sum_{k=1}^\io \F_k(j_{-k},\ldots,j_k),\qquad |\F_k(j_{-k},\ldots,j_k)|\le \f e^{-\l k}\Eqa(A3.5)$$ \0where $\f>0,\l>0$. In particular $h_\pm$ (and $h_0$) enjoy the property \equ(A3.5) ({\it short range}). If $\lis\m_+,\lis\m_-$ are the Gibbs states with formal Hamiltonians: $$\sum_{k=-\io}^\io h_+(\th^k\V j),\qquad \sum_{k=-\io}^\io h_-(\th^k\V j)\Eqa(A3.6)$$ \0the distributions $\m_\pm$ on $ M$, images of $\lis\m_\pm$ via the code $X$ in \equ(A3.2), will be the {\it forward} and {\it backward statistics} of the volume distribution $\m_0$ (corresponding to $\lis\m_0$ via the code $X$), [Si]. This means that: $$\lim_{T\to\io}\fra1T\sum_{k=0}^{T-1} F(S^{\pm k}x)=\ig_ M \m_\pm(dy) F(y)\=\m_\pm (F)\Eqa(A3.7)$$ \0for all smooth $F$ and for $\m_0$--almost all $x\in M$. The distributions $\m_\pm$ are often called the {\it SRB distributions}, [ER]; the above statements and \equ(A3.6),\equ(A3.7) constitute the content of a well known theorem by Sinai, [Si]. An approximation theorem for $\m_+$ can be given in terms of the {\it coarse graining} of $ M$ generated by the markovian pavement $\EE_T=\bigvee_{k=-T}^T S^{-k}\EE$.\annota{3}{\nota Where $\vee$ denotes the operation which, given two pavements $\EE,\EE'$ generates a new pavement $\EE\vee\EE'$: the rectangles of $\EE\vee\EE'$ simply consist of all the intersections $E\cap E'$ of pairs of rectangles $E\in\EE$ and $E'\in\EE'$.} If $E_{j_{-T},\ldots,j_T}\=\cap_{k=-T}^T S^{-k} E_{j_k}$ and $x_{j_{-T},\ldots,j_T}$ is a point chosen in the coarse grain set $E_{j_{-T},\ldots,j_T}$, so that its symbolic sequence is obtained by attaching to the right and to the left of ${j_{-T},\ldots,j_T}$ arbitrary compatible sequences depending only on the symbols $j_{\pm T}$ respectively. We define the distribution $\m_{T,\t}$ by setting: $$\eqalign{\m_{T,\t}(F)\=&\ig_ M \m_{T,\t}(dx) F(x)= \fra{\sum_{j_{-T},\ldots,j_T}\lis\L_{u,\t}^{\,-1} (x_{j_{-T},\ldots,j_T}) F(x_{j_{-T},\ldots,j_T})}{\sum_{j_{-T}, \ldots,j_T}\lis\L_{u,\t}^{\,-1}(x_{j_{-T},\ldots,j_T})}\cr \lis\L_{u,\t}(x){\buildrel def \over =}& \prod_{k=-\t/2}^{\t/2-1}\L_u(S^kx)\cr}\Eqa(A3.8)$$ Then for all smooth $F$ we have: $\lim_{T\ge\t/2,\,\t\to\io} \m_{T,\t}(F)=\m_+(F)$. Note that equation \equ(A3.8) can also be written: $$\m_{T,\t}(F)= \fra{\sum_{j_{-T},\ldots,j_T}e^{-\sum_{k=-\t/2}^{\t/2-1} h_+(\th^k\V j^0)} F(X(\V j^0))} {\sum_{j_{-T}, \ldots,j_T} e^{-\sum_{k=-\t/2}^{\t/2-1} h_+(\th^k\V j^0)}}\Eqa(A3.9)$$ \0where $\V j^0\in C$ is the compatible sequence agreeing with $j_{-T},\ldots,j_T$ between $-T$ and $T$ (\ie $X(\V j^0)=x_{j_{-T},\ldots,j_T}\in E_{j_{-T},\ldots,j_T}$) and continued outside as above. \* \0{\it Notation:} to simplify the notations we shall write, when $T$ is regarded as having a fixed value, $\qq$ for the elements $\qq=(j_{-T},\ldots,j_T)$ of $\{1,\ldots,\NN\}^{2T+1}$; and $E_\qq$ will denote $E_{j_{-T},\ldots,j_T}$ and $x_\qq$ the above point of $E_\qq$. \* \0{\it Remark:} Note that the weights in \equ(A3.9) depend on the special choices of the centers $x_\qq$ (\ie of $\V j^0$); but if $x_\qq$ varies in $E_\qq$ the weight of $x_\qq$ changes by at most a factor, bounded above by some $B<\io$ and below by $B^{-1}$, for all $T\ge0$, and essentially depending only on the symbols corresponding to the sites close to $\pm T$. \* The last formula shows that the forward statistics of $\m_0$ can be regarded as a Gibbs state for a {\it short range one dimensional spin chain with a hard core interaction}. The spin at $k$ is the value of $j_k\in\{1,\ldots,\NN\}$; the short range refers to the fact that the function $h_+(\V j)\=\log \L_u(X(\V j))$, ($\L_u(x)$ {\it being H\"older continuous}), can be represented as in \equ(A3.5) where the $\F_k$ play the role of "many spins" interaction potentials and the hard core refers to the fact that the only spin configurations $\V j$ allowed are those with $M_{j_k,j_{k+1}}\=1$ for all integers $k$. \* \0{\it(B) A Legendre transform.} \* First the function \equ(2.4) is converted to a function on the spin configurations $\V j\in C$: $$\tilde\e_\t(\V j)=\e_\t(X(\V j))=\fra1{\t}\sum_{k=-\t/2}^{\t/2-1} L(\th^k\V j)\Eqa(A3.10)$$ \0where $L(\V j)\=\fra1{\lis\h_+}\log \L^{\,\pm1}(X(\V j))$ has a {\it short range} representation of the type \equ(A3.5). The SRB distribution $\m_+$ is regarded (see above) as a Gibbs state $\lis\m_+$ with short range potential on the space $C$ of the compatible symbolic sequences, associated with a Markov partition $\EE$, [Si], [Ru2]. Therefore, by general large deviations properties of short range Ising systems ([La], [El], [Ol], there is a function $\lis\z(s)$ real analytic in $s$ for $s\in(-p^*,p^*)$ for a suitable $p^*>0$, strictly convex and such that if $p

p-\d-\h'(\t)\cr}\Eqa(A3.13)$$ \0with $\h(\t),\h'(\t)\tende{\t\to\io}0$. \* \0{\it(C) Thermodynamic formalism informations.} \* In this section $X$ will denote a lattice interval, \ie a set of consecutive integers $X=(x,x+1,\ldots,x+n-1)$: hence it should not be confused with the code $X$ of \equ(A3.2). Let $\V j_X=(j_x,j_{x+1},\ldots,j_{x+n-1})$ if $X=(x,{x+1},\ldots,{x+n-1})$ and $n$ is odd, and call $\lis X=x+(n-1)/2$ the {\it center} of $X$. If $\V j\in C$ is an infinite spin configuration we also denote $\V j_X$ the set of the spins with labels $x\in X$. The left shift of the interval $X$ will be denoted by $\th$; \ie by the same symbol of the left shift of a (infinite) spin configuration $\V j$. Let $l_X(\V j_X)=l^{(n)} (j_x,j_{x+1},\ldots,j_{x+n-1})$, and $h^+_X(\V j_X)=h^{(n)}_+(j_x,j_{x+1}, \ldots,j_{x+n-1})$ be translation invariant, \ie functions such that $l_{\th X}(\V j)\= l_X(\V j)$ and $h^+_{\th X}(\V j)$$=h^+_X(\V j)$, and such that the functions $h_+(\V j)$, see \equ(2.4), and $L(\V j)$, see \equ(A3.10), can be written for suitably chosen constants $b_1,b_2,b,b'$: $$\eqalign{ L(\V j)=&\sum_{\lis X= 0}l_X(\V j_X), \qquad h_+(\V j)=\sum_{\lis X= 0}h^+_X(\V j_X)\cr |l_X(\V j_X)|\le& b_1 e^{-b_2 n},\qquad\kern1cm |h^+_X(\V j_X)|\le b e^{-b' n}\cr}\Eqa(A3.14)$$ \0Then $\t\tilde\e_\t (\V j)$ can be written as $\sum_{\lis X\in[-\t/2,\t/2-1]} l_X(\V j_X)$. Hence $\t\tilde\e_\t(\V j)$ can be approximated by $\t\tilde \e_\t^M(\V j) ={\sum}^{(M)} l_X(\V j_X)$ where $\sum^{(M)}$ means summation over the sets $X\subseteq[-\fra12\t-M,\fra12\t+M]$, while $\lis X$ is in $[-\fra12\t,\fra12\t-1]$. The approximation is described by: $$|\t\tilde\e^M_\t(\V j)-\t\tilde\e_\t(\V j)|\le b_3 e^{-b_4 M}\Eqa(A3.15)$$ \0for suitable\annota{4}{\nota One can check from \equ(A3.14), that the constants $b_3,b_4$ can be expressed as simple functions of $b_1,b_2$.} $b_3,b_4$ and for all $M\ge0$. Therefore if $I_{p,\d}=[p-\d,p+\d]$ and $M=0$ we have: $$\m_+(\{\e_\t(x)\in I_{p,\d}\})\cases{\le \lis\m_+(\{\tilde\e^0_\t\in I_{p,\d+b_3/\t}\})\cr \ge \lis\m_+(\{\tilde\e^0_\t\in I_{p,\d-b_3/\t}\})\cr}\Eqa(A3.16)$$ It follows from the general theory of $1$--dimensional Gibbs distributions, [Ru2], that the $\lis\m_+$--pro\-ba\-bi\-li\-ty of a spin configuration which coincides with $\V j_{[-\t/2,\t/2]}$ in the interval $[-\fra12 \t,\fra12\t]$,\annota{5}{\nota \ie the spin configurations $\V j'$ such that $j'_x=j_x$, $x\in[-\fra12\t,\fra12\t]$.} is: $$\fra{\Big[e^{-{\sum}^*h^+_X(\V j_X)}\Big]} {\sum_{\V j'_{[-\t/2,\t/2]}}\Big[\cdot\Big]}\, P(\V j_{[-\t/2,\t/2]})\Eqa(A3.17)$$ \0where $\sum^*$ denotes summation over all the $X\subseteq [-\t/2,\t/2-1]$; the denominator is just the sum of terms like the numerator, evaluated at a generic (compatible) spin configuration $\V j'_{[-\t/2,\t/2]}$; finally $P$ verifies the bound, [Ru2]: $$B_1^{-1}< P(\V j_{\,[-\t/2,\t/2]})< B_1\Eqa(A3.18)$$ \0with $B_1$ a suitable constant independent of $\V j_{\,[-\t/2,\t/2]}$ and of $\t$ ($B_1$ can be explicitly estimated in terms of $b,b'$). Therefore from \equ(A3.16) and \equ(A3.17) we deduce for any $T\ge\t/2$: $$\eqalign{ &\m_+(\{\e_\t(x)\in I_{p,\d}\})\le \lis\m_+(\{\tilde \e^0_\t\in I_{p,\d+b_3/\t}\})\le\cr &\le B_2\,\m_{T,\t}(\{\tilde\e^0_\t\in I_{p,\d+b_3/\t}\})\le B_2\,\m_{T,\t}(\{\tilde\e_\t\in I_{p,\d+2b_3/\t}\})\cr}\Eqa(A3.19)$$ \0for some constant $B_2>0$; and likewise a lower bound is obtained by replacing $B_2$ by $B_2^{-1}$ and $b_3$ by $-b_3$. Then if $p

-\fra{\log B_2^2}{\t\lis\h_+}+\fra1{\t\lis\h_+}\log \fra{\m_{T,\t}(\{\tilde\e_\t\in I_{p,\d}\})}{\m_{T,\t}(\{\tilde\e_\t\in- I_{p,\d}\})}\cr}\Eqa(A3.20)$$ \penalty10000 \0for $I_{p,\d}\subset [-p^*,p^*]$ and for $\t$ so large that $p+\d+2b_3/\t< p^*$.} \* Hence \equ(A3.13) will follow if we can prove: \* \0{\it Lemma 2: there is a constant $\lis b$ such that the approximate SRB distribution $\m_{T,\t}$ verifies: $$\fra1{\lis\h_+\t}\log \fra{\m_{T,\t}(\{\tilde\e_\t\in I_{p,\d}\})}{\m_{T,\t}(\{\tilde\e_\t\in- I_{p,\d}\})}\ \cases{\le p+\d+ \lis b/\t\cr \ge p-\d -\lis b/\t\cr}\Eqa(A3.21)$$ \0for $\t$ large enough (so that $\d+\lis b/\t

-\io$, for all $|s|

\min_\qq
\lis\L^{\,-1}_{u,\t}(x_\qq) \lis\L^{\,-1}_{s,\t}(x_\qq)\cr} \Eqa(A3.23)$$
\0where the maxima are evaluated as $\qq$ varies with $\e_\t(x_\qq)\in
I_{p,\d}$.
\\\hbox{}\kern0.3cm
By \equ(2.1) we can replace
$\lis\L^{\,-1}_{u,\t}(x)\lis\L^{\,-1}_{s,\t}(x)$ with
$\lis\L_\t^{-1}(x)B^{\pm1}$, see \equ(A3.8), \equ(2.4); thus
noting that by definition of the set of $\qq$'s in the maximum
in \equ(A3.23) we have $\fra1{\lis\h_+\t}\log \lis\L^{\,-1}_\t(x_\qq)
\in I_{p,\d}$, we see that \equ(A3.21) follows with $\lis b
=\fra1{\lis\h_+}\log B$.
\*
\0{\it Corollary: the above analysis gives us a concrete bound on the
speed at which the limits in \equ(2.6) are approached. Namely the error
has order $O(\t^{-1})$.}
\*
\0This is so because the limit \equ(A3.11) is reached at speed
$O(\t^{-1})$; furthermore the regularity of $\l(s)$ in \equ(A3.11) and the
size of $\h(\t),\h'(\t)$ and the error term in
\equ(A3.21) have all order $O(\t^{-1})$.
The above analysis proves a large deviation result for the probability
distribution $\m_+$: since $\m_+$ is a Gibbs distribution, see
\equ(A3.6), various other large deviations theorems hold for it, [DV],
[El], [Ol], but unlike the above they are not related to the time reversal
symmetry.
%\ifnum\mgnf=0\pagina\fi
\*
\0{\bf Appendix A4: Heuristic proof of the local fluctuation theorem.}
\numsec=4\numfor=1\*
\0{\it(A) Markov partitions and symbolic dynamics for the chain.}
\*
The reduction of the dynamical nonequilibrium problem of a weakly
interacting chain of Anosov maps, see \S3, to a short range lattice spin
system equilibrium problem is the content of (A), (B) of this appendix,
see [Ga7]. This is an extension of the corresponding analysis in
Appendix A3 for the case of a single Anosov map: it is necessary to
discuss it again in order to exploit the short range nature of the
coupling and its weakness in order to obtain results independent on the
size $N$ of the chain.
Let $\lis \PP_0=(E^0_1,\ldots,E^0_{\NN_0})$ be a Markov partition, see
[Si], for the unperturbed ``single site'' system $(\lis M_0\times \lis
M_0, \lis S_0\times \lis S_0^{\,-1})$. Then
$\lis\PP_0^{2N+1}=\{E_\a\}$, $\a=(\r_{-N},\ldots,\r_N)$ with $E_\a=
E^0_{\r_{-N}}\times E^0_{\r_{-N+1}}\times\ldots \times E^0_{\r_N}$ is
a Markov partition of $(\lis M_0^{2(2N+1)}, S_0)$.
The perturbation, {\it if small enough}, will deform the partition
$\lis\PP_0^{2N+1}$ into a Markov partition $\PP$ for $(M,S)$ changing
only ``slightly'' the partition $\lis\PP_0^{2N+1}$. The work [PS]
shows that the above ``$\e$ small enough'' {\it mean that $\e$ has to
be chosen small but that it can be chosen $N$--independent}, as we
shall always suppose in what follows.
Under such circumstances we can establish a correspondence between
points of $M$ that have the same ``symbolic history'' (or ``symbolic
dynamics'') along $\lis\PP_0^{2N+1}$ under $S_0$ and along $\PP$ under
$S$; we shall denote it by $h$; see [PS].
The Markov partition $\lis\PP_0^{2N+1}$ for $S_0$ associates with each
point $\xx=(x_{-N},\ldots, x_N)$ a sequence $(\s_{i,j})$, $i\in
[-N,N], j\in (-\io,\io)$ of symbols so that $(\s_{i,j})_{j=-\io}^\io$
is the free symbolic dynamics of the point $x_i$. We call the first
label $i$ of $\s_{i,j}$ a ``space--label'' and the second a
``time--label''. Not all sequences can arise as histories of points:
however (by the definition of $h$, see above) precisely the same
sequences arise as histories of points along $\PP_0$ under the free
evolution $S_0$ or along $\PP$ under the interacting evolution $S$.
The map $h$ is H\"older continuous and ``short ranged'':
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$$|h(\xx)_i-h(\xx')_i|\le C\sum_j \e^{|i-j|\g'} |x_j-x'_j|^\g\Eqa(A4.1)$$
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for some $\g,\g',C>0$, [PS], if $|x-y|$ denotes the distance in $\lis
M_0\times\lis M_0$ (\ie in the single site phase space).
Furthermore the code $\xx\otto\V\s$ associating with $\xx$ its
``history'' or ``symbolic dynamics'' $\V\s(\xx)$ along the partition
$\PP$ under the map $S$ is such that, fixed $j$:
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$$\V\s(\xx)_i=\V\s(\xx')_i\ {\rm for}\ |i-j|\le\ell \qquad \tto\qquad
|x_j-x'_j| \le C\e^{\g \ell}\Eqa(A4.2)$$
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The inverse code associating with a history $\V\s$ a point with such history
will be denoted $\xx(\V\s)$.
If $\xx=(x_{-N},\ldots, x_N)$ is coded into
$\V\s(\xx)=(\V\s_{-N},\ldots,\V\s_N)=(\s_{i,j})$, with
$i=-N,\ldots,N$, and $j\in (-\io,+\io)$, the short range property
holds also in the time direction. This means that, fixed $i_0$:
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$$\s_{i,j}=\s'_{i,j}\ {\rm for}\ |i-i_0|