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In discrete schemes, weak KAM solutions may be interpreted as approximations of correctors for some Hamilton-Jacobi equations in the periodic setting. It is known that correctors may not exist in the almost periodic setting. We show the existence of discrete weak KAM solutions for non-degenerate and weakly twist interactions in general. Furthermore, assuming equivariance with respect to a linearly repetitive quasi-periodic set, we completely classify all possible types of weak KAM solutions., Comment: 53 pages, 7 figures

In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the collision cross-section $\mathcal{B}$, we prove that the linearized Boltzmann operator $\mathcal{L}$ of this model is a Fredholm operator. For this, we write $\mathcal{L}$ as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator $\mathcal{K}$ is compact. The result is established after inspecting the kernel form of $\mathcal{K}$ and proving it to be $L^2$ integrable over its domain using elementary arguments.

We construct a finite-range potential on a bidimensional full shift on a finite alphabet that exhibits a zero-temperature chaotic behavior as introduced by van Enter and Ruszel. This is the phenomenon where there exists a sequence of temperatures that converges to zero for which the whole set of equilibrium measures at these given temperatures oscillates between two sets of ground states. Br\'emont's work shows that the phenomenon of non-convergence does not exist for finite-range potentials in dimension one for finite alphabets; Leplaideur obtained a different proof for the same fact. Chazottes and Hochman provided the first example of non-convergence in higher dimensions $d\geq3$; we extend their result for $d=2$ and highlight the importance of two estimates of recursive nature that are crucial for this proof: the relative complexity and the reconstruction function of an extension. We note that a different proof of this result was found by Chazottes and Shinoda, at around the same time that this article was initially submitted and that a strong generalization has been found by Gayral, Sablik and Taati., Comment: Final version. Accepted for publication in Advances in Mathematics

Liv\v{s}ic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Liv\v{s}ic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a H\"older coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory., Comment: 32 pages,2 picture, this is a flow version of the paper "Lipschitz sub-actions for locally maximal hyperbolic sets of a C1 map" published at Discrete and Continuous Dynamical Systems working with (Anosov) maps

Liv\v{s}ic theorem asserts that, for Anosov diffeomorphisms/flows, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Liv\v{s}ic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not necessarily bijective nor transitive. We actually prove our main result in the setting of locally maximal hyperbolic sets for not general $C^1$ map. The construction of the coboundary uses a new notion of the Lax-Oleinik operator that is a standard tool in the discrete Aubry-Mather theory., Comment: 30 pages, 2 figures

Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the discounted discrete weak KAM solution converges to some selected weak KAM solution. We show that, in the ergodic optimization case for a coboundary observable over a minimal system, the discounted transfer function converges if and only if the observable is balanced., Comment: 11 pages

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index $d$ admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest $d$-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension $d$ and a slow space of co-dimension $d$. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite-dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved, and no compactness of the underlying system or smoothness of the cocycle is required. Moreover, we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces.

We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols $\{0,1\}$. These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are locally constant, Lipschitz continuous or, more generally, of summable variation. We assume there exists exactly two ground states: the fixed points $0^\infty$ and $1^\infty$. We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero., Comment: 27 pages, 2 figures. To appear in Ergodic Theory and Dynamical Systems

We develop two approximation schemes for solving the cell equation and the discounted cell equation using Aubry-Mather-Fathi theory. The Hamiltonian is supposed to be Tonelli, time-independent , and periodic in space. By Legendre transform it is equivalent to find a fixed point of some nonlinear operator, called Lax-Oleinik operator, which may be discounted or not. By discretizing in time, we are led to solve an additive eigenvalue problem involving a discrete Lax-Oleinik operator. We show how to approximate the effective Hamiltonian and some weak KAM solutions by letting the time step in the discrete model tend to zero. We also obtain a selected discrete weak KAM solution as in [Davini et al 2014] and show it converges to a particular solution of the cell equation. In order to unify the two settings, continuous and discrete , we develop a more general formalism of short-range interactions., Comment: 36 pages

The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we show the existence of these configurations for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure, for any environment, the existence of calibrated configurations when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone sets

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice 1,...,d}^N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator $L={\mc L}_A -I$, where \mc L_A is a discrete time Ruelle operator (transfer operator), and A:{1,...,d}^N \to R is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the\emph{a priori}probability. Given a Lipschitz interaction V:\{1,...,d\}^{\bb N}\to \mathbb{R}, we are interested in Gibbs (equilibrium) state for such $V$. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative, and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer., Comment: to appear Journ. of Stat. Physics

Consider the shift $\sigma$ acting on the Bernoulli space $\Sigma={1,2,...,n}^\mathbb{N}$. We denote $\hat{\Sigma}= {1,2,...,n}^\mathbb{Z}$. We analyze several properties of the maximizing probability $\mu_{\infty,A}$ of a Holder potential $A: \Sigma \to \mathbb{R}$. Associated to $A(x)$, via the involution kernel, $W: \hat{\Sigma} \to \mathbb{R}$, it is known that can we get the dual potential $A^*(y)$, where $(x,y)\in \hat{\Sigma}$. Consider $\mu_{\infty, A^*}$ a maximizing probability for $A^*$. We would like to consider the transport problem from $\mu_{\infty,A}$ to $\mu_{\infty,A^*}$. In this case, it is natural to consider the cost function $c(x,y) = I(x) - W(x,y) +\gamma $, where $I$ is the deviation function. The pair of functions for the Kantorovich Transport dual Problem are $(-V,-V^*$), where we denote the two calibrated sub-actions by $V$ and $V^*$, respectively, for $A$ and $A^*$ for $\mu_{\infty,A}$. We analyze the graph property for the optimal plan $\hat{\mu}$., Comment: to appear in Mathematics of Planet Earth, Volume 1 Dynamics, Games and Science. Springer Verlag, Edit. Alberto Pinto et all

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e 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 $\partial\DD=\SS^1$. The geodesic flow on the compact surface $\DD/\Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:\SS^1\to\SS^1$, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. 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 \[ \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \] \noindent 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 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 only 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, $\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $(x,y) \in [0,1]^2$, we show the graph property.

Consider a transitive expanding dynamical system $ \sigma: \Sigma \to \Sigma $, and a H\"older potential $ A $. In ergodic optimization, one is interested in properties of $A$-maximizing probabilities. Assuming ergodicity, it is already known that the projection of the support of such probabilities is contained in the set of non-wandering points with respect to $ A $, denoted by $ \Omega(A) $. A separating sub-action is a sub-action such that the sub-cohomological equation becomes an identity just on $ \Omega(A) $. For a fixed H\"older potential $ A $, we prove not only that there exists H\"older separating sub-actions but in fact that they define a residual subset of the H\"older sub-actions. We use the existence of such separating sub-actions in an application for the case one has more than one maximizing probability. Suppose we have a finite number of distinct $A$-maximizing probabilities with ergodic property: $ \hat \mu_j $, $ j \in \{1, 2, ..., l\} $. Considering a calibrated sub-action $ u $, under certain conditions, we will show that it can be written in the form $$ u (\mathbf x)= u (\mathbf x^i) + h_A(\mathbf x^i, \mathbf x), $$ for all $ \mathbf x \in \Sigma $, where $ \mathbf x^i $ is a special point (in the projection of the support of a certain $ \hat \mu_i $) and $ h_A $ is the Peierls barrier associated to $ A $.