Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams

Ergodic properties of equilibrium measures for smooth three dimensional flows

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $\mu$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.Comment: 32 pages, 1 figure, a section on equilibrium measures for multiples of the geometric potential has been added, to appear in Commentarii Mathematici Helvetic

No temporal distributional limit theorem for a.e. irrational translation

Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero

Local limit theorems for inhomogeneous Markov chains

We prove the Local Limit Theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov arrays. As an application we obtain the precise asymptotics in the large deviation regime for bounded additive functionals of uniformly elliptic Markov chains. The proofs rely on new reduction theorems for Markov arrays.Comment: 212 page

Continuity properties of Lyapunov exponents for surface diffeomorphisms

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy