Countable Markov Partitions Suitable for Thermodynamic Formalism

TL;DR

This paper constructs new countable Markov partitions for hyperbolic attractors with stronger contraction than expansion, enabling thermodynamic formalism and proving exponential decay of correlations for certain functions.

Contribution

It introduces a novel method for creating Markov partitions with Cantor set cross-sections, facilitating thermodynamic analysis of hyperbolic attractors.

Findings

Existence of Markov partitions with positive Lebesgue measure cross-sections

Development of thermodynamic formalism for these partitions

Proof of exponential decay of correlations for Hölder functions

Abstract

We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion we construct new Markov rectangles such that their crossections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain H\"older functions. The results are based on the methods developed by Sarig.