Decay of Correlations for some Non-Uniformly Hyperbolic Attractors

Abstract

A classical problem in smooth dynamical systems is known as smooth realization problem. It asks if given a compact manifold $M$, one can construct a volume preserving diffeomorphism with prescribed ergodic properties. We study the decay of correlations for certain dynamical systems with non-uniformly hyperbolic attractors, which natural invariant measure is not the volume, but the SRB measure. The system $g$ that we consider is produced by applying the slow-down procedure to a uniformly hyperbolic diffeomorphism $f$ with an attractor. Under certain assumptions on the map $f$ and the slow-down neighborhood, we show that the map $g$ admits polynomial upper and lower bounds on correlations with respect to its SRB measure and the class of H\"older continuous potentials. Our results apply to the Smale-Williams solenoid, as well as its sufficiently small perturbations.