Anosov diffeomorphisms and coupling

Abstract

We prove the existence of a Sinai–Ruelle–Bowen (SRB) measure and the exponential decay of correlations for smooth observables for mixing Anosov C^{1+\alpha} diffeormorphisms on a d-dimensional (d \geq 2) Riemannian manifold. The novelty lies in the very simple method of proof. We construct explicitly a coupling between two initial densities so that under the action of the diffeomorphism, both components get closer and closer. The speed of this convergence can be explicitly estimated and is directly related to the speed of decay of the correlations. The existence of the SRB measure and its properties readily follow.