A relation between entropy and transitivity of Anosov diffeomorphisms

Abstract

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Transitivity of Anosov diffeomorphisms can be reached under suitable hypotheses on Lyapunov exponents on the set of periodic points and the structure of the MME. In another way, assuming together the uniqueness of MME and that every point is regular, in Oseledec’s Theorem sense, also we can get the transitivity of Anosov diffeomorphisms in this setting.