Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities

TL;DR

This paper constructs a Markov partition for 2D hyperbolic systems with singularities, enabling the derivation of statistical properties and thermodynamic formalism, applicable to Sinai billiards and their perturbations.

Contribution

It introduces a countable Markov partition for hyperbolic systems with singularities, facilitating analysis of their statistical and thermodynamic properties.

Findings

Established decay rates of correlations.

Proved the central limit theorem for the systems.

Applied results to Sinai dispersing billiards and perturbations.

Abstract

For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentials, by using the inducing scheme of hyperbolic type. All the results apply to Sinai dispersing billiards, and their small perturbations due to external forces and nonelastic reflections with kicks and slips.