A relation between entropy and transitivity of Anosov diffeomorphisms

TL;DR

This paper explores the relationship between entropy and transitivity in Anosov diffeomorphisms, showing conditions under which entropy properties imply transitivity.

Contribution

It establishes conditions involving Lyapunov exponents and measure structure that ensure transitivity of $C^1$-Anosov diffeomorphisms, complementing known results about entropy.

Findings

Transitivity can be inferred from entropy and Lyapunov exponent conditions.

Conditions on the measure of maximal entropy lead to transitivity.

The paper provides a converse perspective to known entropy-transitivity relations.

Abstract

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Under suitable hypothesis on Lyapunov exponents on the set of periodic points and the structure of the MME we get transitivity of $C^1-$Anosov diffeomorphisms.