Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

Abstract

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

SRB measures of singular hyperbolic attractors

“…These are invariant measures for hyperbolic dynamical systems that have conditional measures on unstable leaves that are absolutely continuous with respect to the Riemannian leaf volume. For uniformly hyperbolic dynamical systems (such as transitive Anosov diffeomorphisms and attractors of Axiom A systems), there is a unique SRB measure [9], and the existence of SRB measures has been established for several classes of nonuniformly hyperbolic dynamical systems [10,14,18] and partially hyperbolic dynamical systems [2,8]. It was further shown in [15] that if M is a compact Riemannian 2-manifold and f : M → M is a hyperbolic diffeomorphism admitting an SRB measure, then this SRB measure is unique.…”