Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows

TL;DR

This paper develops a symbolic coding method for three-dimensional flows with positive speed, capturing a broad class of hyperbolic measures and orbits, enabling detailed analysis of their dynamical properties.

Contribution

It introduces a new symbolic dynamics construction for large non-uniformly hyperbolic sets in 3D flows, extending previous measure-specific codings to broader hyperbolic sets.

Findings

Codes a set of full measure for each hyperbolic measure

Includes all ergodic measures with entropy > χ

Codes hyperbolic periodic orbits with Lyapunov exponents outside [-χ, χ]

Abstract

We construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each $\chi>0$, we code a set of full measure for every invariant probability measure which is $\chi$-hyperbolic. These include all ergodic measures with entropy bigger than $\chi$ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of $[-\chi,\chi]$. This contrasts with a previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts.