Entropy of geometric structures

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic properties for this entropy, including the \emph{additivity property}, analogous to the additivity of Clausius--Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves.Comment: The results of this paper were announced in a talk last year in IMPA, Rio (Poisson 2010

Phase transition and correlation decay in Coupled Map Lattices

For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya's probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling. This implies the exponential decay, in space and in time, of the correlation functions of the invariant measures

On the Integrability of Intermediate Distributions for Anosov Diffeomorphisms

. We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of C 1 \GammaAnosov diffeomorphism on three dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant. 1. Introduction Let F be a C r \Gammadiffeomorphism of a compact smooth Riemannian manifold M , r = 1; 2; \Delta \Delta \Delta ; 1 and \Gamma 0 (TM) the space of C 0 \Gammavector fields on M . The map F induces an invertible bounded linear operator on \Gamma 0 (TM) by: F v(x) = DFv(F \Gamma1 (x)); v(x) 2 \Gamma 0 (TM); x 2 M: The Mather spectrum oe(F ) is defined to be the spectrum of the complexification of F . In [M], Mather showed that F is an Anosov system, if and only if 1 = 2 oe(F ), and moreover, if the nonperiodic points of F are dense in M ..