Maximizing measures for partially hyperbolic systems with compact center leaves

TL;DR

This paper establishes a dichotomy for accessible partially hyperbolic diffeomorphisms with compact center leaves on 3-manifolds, characterizing entropy maximizing measures and their properties, with implications for topological conjugacy and system mixing.

Contribution

It provides a new dichotomy classification for entropy maximizing measures in these systems, linking measure properties to topological conjugacy and genericity.

Findings

Unique entropy maximizing measure with Bernoulli property and zero center Lyapunov exponent

Finite entropy maximizing measures with nonzero exponents, including both positive and negative

Existence of open sets of topologically mixing diffeomorphisms with multiple entropy maximizing measures

Abstract

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of 3-dimensional manifolds having compact center leaves: either there is a unique entropy maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0 or, there is a finite number of entropy maximizing measures, all of them with nonzero center Lyapunov exponent (at least one with negative exponent and one with positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence we obtain an open set of topologically mixing diffeomorphisms having more than one entropy maximizing measure.