Symbolic dynamics for non uniformly hyperbolic diffeomorphisms of compact smooth manifolds

TL;DR

This paper develops symbolic coding for non-uniformly hyperbolic diffeomorphisms on compact manifolds, enabling detailed analysis of hyperbolic measures and periodic orbits across various dimensions.

Contribution

It extends Sarig's surface-based work to higher dimensions by constructing countable Markov partitions for non-uniform hyperbolic diffeomorphisms on compact manifolds.

Findings

Countable Markov partitions for non-uniform hyperbolic diffeomorphisms in any dimension.

Symbolic coding for hyperbolic measures with Lyapunov exponents bounded away from zero.

Results on counting hyperbolic periodic orbits and structure of measures of maximal entropy.

Abstract

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.