Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

TL;DR

This paper proves exponential decay of correlations for certain contracting fiber maps and applies these results to establish logarithm laws for singular hyperbolic flows, enhancing understanding of their statistical properties.

Contribution

It introduces a method to derive exponential decay of correlations for maps with contracting fibers and applies it to singular hyperbolic flows to establish logarithm laws.

Findings

Exponential decay of correlations for a class of maps with contracting fibers.

Logarithm laws for singular hyperbolic flows.

Application of decay results to Poincaré maps of flows.

Abstract

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.