Thermodynamics of smooth models of pseudo-Anosov homeomorphisms

TL;DR

This paper develops a thermodynamic formalism for smooth models of pseudo-Anosov homeomorphisms, establishing the existence and uniqueness of equilibrium states, including SRB measures and measures of maximal entropy, with strong statistical properties.

Contribution

It introduces a new smooth realization of pseudo-Anosov maps with fixed singularities and applies Young towers to prove key thermodynamic results.

Findings

Existence and uniqueness of equilibrium states for geometric t-potentials.

Identification of a measure of maximal entropy with exponential decay of correlations.

Establishment of the Central Limit Theorem for the measures studied.

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.