Constant periodic data and entropy of Anosov diffeomorphisms

TL;DR

This paper investigates how the constant periodic data condition influences the topological entropy of Anosov diffeomorphisms, revealing finite measures of maximal entropy and characterizing transitivity.

Contribution

It establishes that under constant periodic data, Anosov diffeomorphisms have finitely many maximal entropy measures, all absolutely continuous, and links transitivity to these measures.

Findings

Finitely many measures of maximal entropy exist under constant periodic data.

All maximal entropy measures are absolutely continuous with respect to Lebesgue.

Transitivity is characterized via measures of maximal entropy in smooth Anosov diffeomorphisms.

Abstract

We study the effects that the constant periodic data condition have on topological entropy of Anosov diffeomorphisms. Under constant periodic data condition we prove that Anosov diffeomorphism has finitely many measures of maximal entropy and each one of them is absolutely continuous with respect to Lebesgue. From this, in the setting of $C^{\infty}-$Anosov diffeomorphisms satisfying constant periodic data, we provide a characterization of transitivity property via measures of maximal entropy.