Entropy and variational principles for holonomic probabilities of IFS
Artur O. Lopes Elismar R. Oliveira
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 937-955 doi: 10.3934/dcds.2009.23.937
An IFS ( iterated function system), $([0,1], \tau_{i})$, on the interval $[0,1]$, is a family of continuous functions $\tau_{0},\tau_{1}, ..., \tau_{d-1} : [0,1] \to [0,1]$. Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$, defined by $\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$ where $\Sigma=\{0,1, ..., d-1\}^{\mathbb{N}}$, $\sigma: \Sigma \to \Sigma$ 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 a probability $\nu $ on $[0,1]$ satisfying $ P_{u}(h)=\rho h, \quad P_{u}^{*}(\nu)=\rho \nu$.
A probability $\hat{\nu}$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma}$, if, $ \int\, g \circ \hat{\sigma}\, d\hat{\nu}= \int \,g \,d\hat{\nu}, \, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal H$.
For a holonomic probability $\hat{\nu}$ on $[0,1]\times \Sigma$ we define the entropy $h(\hat{\nu})$=inf$_f \in \mathbb{B}^{+} \int \ln(\frac{P_{\psi}f}{\psi f}) d\hat{\nu}\geq 0$, where, $\psi \in \mathbb{B}^{+}$ is a fixed (any one) positive potential.
Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find solutions of the maximization problem $p(\phi)$= sup$_\hat{\nu} \in \mathcal{H} \{ \,h(\hat{\nu}) + \int \ln(\phi) d\hat{\nu} \,\}.$ We show an example where a holonomic not-$\hat{\sigma}$-invariant probability attains the supremum value. In the last section we consider maximizing probabilities, sub-actions and duality for potentials $A(x,w)$, $(x,w)\in [0,1]\times \Sigma$, for IFS.
keywords: entropy pressure holonomic probability Iterated Function System maximizing probability
Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice
Leandro Cioletti Artur O. Lopes
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6139-6152 doi: 10.3934/dcds.2017264

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these ''in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures).

Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space -we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

keywords: Thermodynamic formalism Ruelle operator equilibrium states DLR-Gibbs measures
Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics
Artur O. Lopes Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1155-1174 doi: 10.3934/dcds.2011.29.1155
We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.
keywords: twisted Brownian motion. large deviation Geodesic flow Aubry-Mather measure
Cohomology and subcohomology problems for expansive, non Anosov geodesic flows
Artur O. Lopes Vladimir A. Rosas Rafael O. Ruggiero
Discrete & Continuous Dynamical Systems - A 2007, 17(2): 403-422 doi: 10.3934/dcds.2007.17.403
We show that there are examples of expansive, non-Anosov geodesic flows of compact surfaces with non-positive curvature, where the Livsic Theorem holds in its classical (continuous, Hölder) version. We also show that such flows have continuous subaction functions associated to Hölder continuous observables.
keywords: action functions Livsic Theorem subaction functions Expansive geodesic flow cohomology of dynamical systems.
Selection of calibrated subaction when temperature goes to zero in the discounted problem
Renato Iturriaga Artur O. Lopes Jairo K. Mengue
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 4997-5010 doi: 10.3934/dcds.2018218

Consider $T(x) = d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $zhongwenzy<\lambda<1$ and the unique function $b_\lambda:S^1 \to \mathbb{R}$ satisfying $ b_\lambda(x) = \max_{T(y) = x} \{ \lambda \, b_\lambda(y) + A(y)\}. $

We will show that, when $\lambda \to 1$, the function $b_\lambda- \frac{m(A)}{1-\lambda}$ converges uniformly to the calibrated subaction $V(x) = \max_{\mu \in \mathcal{ M}} \int S(y, x) \, d \mu(y)$, where $S$ is the Mañe potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$.

For $\beta>0$ and $\lambda \in (0, 1)$, there exists a unique fixed point $u_{\lambda, \beta} :S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$ converges uniformly to the main eigenfuntion $\phi_\beta $ for the Ruelle operator associated to $\beta A$. We consider $\lambda = \lambda(\beta)$, $\beta(1-\lambda(\beta))\to+\infty$ and $\lambda(\beta) \to 1$, as $\beta \to\infty$. Under these hypotheses we will show that $\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$.

keywords: Calibrated subaction maximizing measure discounted method zero temperature limit eigenfunction of the Ruelle operator

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