Measures of maximal entropy for monotone factors of smooth surface diffeomorphisms

Abstract

We prove that a transitive monotone factor of a smooth surface diffeomorphism has a unique measure of maximal entropy whenever its topological entropy is positive. Its entropy is the entropy of a single homoclinic class of the diffeomorphism, and the factor map is a measure-theoretic isomorphism on every ergodic measure with positive entropy of that class. The other homoclinic classes are mapped to periodic orbits. We give an application to the essential factor of a diffeomorphism isotopic to a pseudo-Anosov map ---possibly relative to a finite invariant set--- constructed by García-Sassi \& Tal: the essential factor has a unique measure of maximal entropy, which is Bernoulli and has full support.