Chaotic temperature dependence at zero temperature

We present a class of examples of nearest-neighbour, boubded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension

Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to H\"older continuous potentials.Comment: 33 pages, 6 figure

Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Holder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.Comment: 29 pages, Annales de l'Institut Henri Poincare - Analyse non lineaire (to appear

Flatness is a Criterion for Selection of Maximizing Measures

For a full shift with Np+1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $\sigma$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $\Pi$ the set of all Borel probabilities $\pi$ on $X\times X$, such that, both its $x$ and $y$ marginal are $\sigma$-invariant probabilities. We are interested in the optimal plan $\pi$ which minimizes $\int c d \pi$ among the probabilities on $\Pi$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $\pi$ by convex combinations of plans such that the support projects in periodic orbits

Substreetutions and more on trees

V2. References to invariant measures on trees added (Kindly given by K. Petersen)We define a notion of substitution on colored binary trees that we call substreetution. We show that a fixed point by a substreetution may be (or not) almost periodic, thus the closure of the orbit under $\mathbb{F}_2^+$-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measure on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk

La sécurité alimentaire à l'heure du néo-libéralism : 3. Les initiatives de substitution

L'étude de l'approvisionnement vivrier de Brazzaville est révélatrice de l'aptitude de certaines sociétés rurales africaines à répondre efficacement aux fortes demandes urbaines. On remarque ici une atomisation des acteurs du commerce qui permet de couvrir un secteur productif lui-même atomisé. En période de crise, les marges sont ainsi redistribuées à un grand nombre de ménages. Outre le fait de remettre en question certaines analyses alarmistes, cette recherche a permis d'insister sur les formes de structuration d'un secteur vigoureux trop souvent dédaigné, voire perturbé par l'Etat. Ce dernier préfère en effet investir dans des projets, a priori certifiés plus efficaces car plus modernes, plutôt que d'encourager les dynamiques endogènes. (Résumé d'auteur