Decay of correlations for some partially hyperbolic diffeomorphisms (Q1026991)

In this paper, \(M\) is a \(d\)-dimensional manifold, with \(d\geq 2\), and \(f\) is a diffeomorphism of \(M\). A probability measure \(\mu\) which is \(f\)-invariant is said to be Ruelle-Sinai-Bowen (RSB) if (i) \(\mu\) has positive Lyapunov exponents and (ii) \(\mu\) has absolutely continuous conditional measures on unstable manifolds. The existence of SRB measures with exponential decay of correlations for Hölder continuous functions was previously studied for Hénon maps, and for some partially hyperbolic diffeomorphisms. The main result of the paper under review is that there exist \(C^{1+\alpha}\)-almost Anosov diffeomorphisms \(f\) with uniformly contracting direction such that \(f\) admits a unique SRB measure with polynomial upper bounds on correlations.