Thermodynamical $u$-formalism I: measures of maximal $u$-entropy for maps that factor over Anosov

TL;DR

This paper develops a method to construct measures of maximal $u$-entropy for certain partially hyperbolic diffeomorphisms, revealing a finite-dimensional structure and ergodic measures with disjoint supports.

Contribution

It introduces a new approach to measure construction for maps factoring over Anosov automorphisms with mostly contracting centers, highlighting the structure of the measure space.

Findings

Finite-dimensional space of maximal $u$-entropy measures

Extreme points are ergodic with disjoint supports

Applicable to partially hyperbolic diffeomorphisms over Anosov torus automorphisms

Abstract

We construct measures of maximal $u$-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has a finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.