The structure of the space of ergodic measures of transitive partially hyperbolic sets

TL;DR

This paper explores the complex structure of ergodic measures in transitive partially hyperbolic systems, revealing a division into exposed and core parts with distinct homoclinic properties and discussing measures of maximal entropy, including nonhyperbolic cases.

Contribution

It introduces new examples of homoclinic classes blending hyperbolic types and analyzes the topology of ergodic measures, including nonhyperbolic measures of maximal entropy.

Findings

Existence of homoclinic classes with disjoint exposed and core parts.

Description of the topology of ergodic measures supported on these parts.

Examples of nonhyperbolic measures of maximal entropy, including unique nontransitive cases.

Abstract

We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an "exposed" piece which is poorly homoclinically related with the rest and a "core" with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic measures. Measures of maximal entropy are discussed in more detail. We present examples where the measure of maximal entropy is nonhyperbolic. We also present examples where the measure of maximal entropy is unique and nonhyperbolic, however in this case the dynamics is nontransitive.