Diffeomorphisms admitting SRB measures and their regularity

Abstract

We are interested in the stochastic property of some "Anosov-like" system. In this paper we will treat a transitive and partially hyperbolic diffeomorphism f of a 2-dimensional torus with uniformly contracting direction, and show that if f is of C² and admits an SRB measure, then f is an Anosov diffeomorphism. In our proof we use the Pujals-Sambarino theorem for C² diffeomorphisms with dominated splitting. In the case of C^1+α the above statement is not true in general, i.e. we can construct a C^1+α counter example of Maneville-Pomeau type.