We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1
051-1082 (2004). We show only the uniqueness of the beta-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $mu$) and its relation with the eigen-probability $ u_beta$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state.
We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set
$S$ and a stationary continuous time Markov Chain $X_t$, $tgeq 0$, taking values on S. We denote by $Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Omega, {cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${cal L}^infty (P)$. From the probability $P$ we define a Ruelle operator ${cal L}^t, tgeq 0$, acting on functions $f:Omega to mathbb{R}$ of ${cal L}^infty (P)$. Given $V:Omega to mathbb{R}$, such that is constant in sets of the form ${X_0=c}$, we define a modified Ruelle operator $tilde{{cal L}}_V^t, tgeq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ u_V$ on $Omega$ associated to $tilde{{cal L}}^t_V, tgeq 0$. We also show the following property for the probability $ u_V$: for any integrable $gin {cal L}^infty (P)$ and any real and positive $t$ $$ int e^{-int_0^t (V circ Theta_s)(.) ds} [ (tilde{{cal L}}^t_V (g)) circ theta_t ] d u_V = int g d u_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).
We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^mathbb{N}to mathbb{R}$ which depends o
nly on the two first coordinates. We are interested in finding stationary Markov probabilities $mu_infty$ on $ [0,1]^mathbb{N}$ that maximize the value $ int A d mu,$ among all stationary Markov probabilities $mu$ on $[0,1]^mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $mu_beta$ which weakly converges to $mu_infty$. The probabilities $mu_beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $frac{partial^2 A}{partial_x partial_y} (x,y) eq 0$, for all $(x,y) in [0,1]^2$, we show the graph property.
We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $mu$ is unique and the support of $mu$ is the Aubry set). Consider, for each value of $epsilon $ and h,
the entropy penalized Mather problem $min {int_{tntimesrn} L(x,v)dmu(x,v)+epsilon S[mu]},$ where the entropy S is given by $S[mu]=int_{tntimesrn}mu(x,v)lnfrac{mu(x,v)}{int_{rn}mu(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $mu_{epsilon, h}$ which converges to the Mather measure $mu$. We show a LDP $lim_{epsilon,hto0} epsilon ln mu_{epsilon,h}(A),$ where $Asubset mathbb{T}^Ntimesmathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+ ablaphi_0(x)(v)-bar{H}_{0},$ where $phi_0$ is the unique viscosity solution for L.
Let $Gamma$ be a co-compact Fuchsian group of isometries on the Poincare disk $DD$ and $Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $Delta$, equivariant by $Gamma$ with real eigenvalue $lambda=-s(1-s)$, where
$s={1/2}+ it$, admits an integral representation by a distribution $dd_{f,s}$ (the Helgason distribution) which is equivariant by $Gamma$ and supported at infinity $partialDD=SS^1$. The geodesic flow on the compact surface $DD/Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:SS^1toSS^1$, the so-called Bowen-Series transformation. Let $ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-sln |T|$. M. Pollicott showed that $dd_{f,s}$ is an eigenfunction of the dual operator $ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $psi_{f,s}$ of $ll_s$ for the eigenvalue 1, given by an integral formula [ psi_{f,s} (xi)=int frac{J(xi,eta)}{|xi-eta|^{2s}} dd_{f,s} (deta), ] oindent where $J(xi,eta)$ is a ${0,1}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $DD/Gamma$.
We show that for a fixed curve $K$ and for a family of variables curves $L$, the number of $n$-Poncelet pairs is $frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd $ (m,n)=1$. The curvee $K$ d
o not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion .
Associated to a IFS one can consider a continuous map $hat{sigma} : [0,1]times Sigma to [0,1]times Sigma$, defined by $hat{sigma}(x,w)=(tau_{X_{1}(w)}(x), sigma(w))$ were $Sigma={0,1, ..., d-1}^{mathbb{N}}$, $sigma: Sigma to Sigma$ is given by$sigma(
w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : Sigma to {0,1, ..., n-1}$ is the projection on the coordinate $k$. A $rho$-weighted system, $rho geq 0$, is a weighted system $([0,1], tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] to mathbb{R}$ and probability $ u $ on $[0,1]$ satisfying $ P_{u}(h)=rho h, quad P_{u}^{*}( u)=rho u$. A probability $hat{ u}$ on $[0,1]times Sigma$ is called holonomic for $hat{sigma}$ if $ int g circ hat{sigma} dhat{ u}= int g dhat{ u}, forall g in C([0,1])$. We denote the set of holonomic probabilities by ${cal H}$. Via disintegration, holonomic probabilities $hat{ u}$ on $[0,1]times Sigma$ are naturally associated to a $rho$-weighted system. More precisely, there exist a probability $ u$ on $[0,1]$ and $u_i, iin{0, 1,2,..,d-1}$ on $[0,1]$, such that is $P_{u}^*( u)= u$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $phi in mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(phi)=$$
In this work we study a class of stochastic processes ${X_t}_{tinN}$, where $X_t = (phi circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability mu_s on [0,1] and a continuous by part function
$phi:[0,1] to R$. We consider here $T_s:[0,1]to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory.
In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):Tt^ntimesRr^nto Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + < P,
v>. For each value of $epsilon $ and $h$, consider the operator Gg[phi](x):= -epsilon h {ln}[int_{re^N} e ^{-frac{hL(x,v)+phi(x+hv)}{epsilon h}}dv], as well as the reversed operator bar Gg[phi](x):= -epsilon h {ln}[int_{re^N} e^{-frac{hL(x+hv,-v)+phi(x+hv)}{epsilon h}}dv], both acting on continuous functions $phi:Tt^nto Rr$. Denote by $phi_{epsilon,h} $ the solution of $Gg[phi_{epsilon,h}]=phi_{epsilon,h}+lambda_{epsilon,h}$, and by $bar phi_{epsilon,h} $ the solution of $bar Gg[phi_{epsilon,h}]=bar phi_{epsilon,h}+lambda_{epsilon,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {cal L} (phi) (x) = int e^{- frac{h L (x,v)}{epsilon}} phi(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.
