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

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.