Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous

TL;DR

This paper demonstrates that on any surface, one can construct area-preserving diffeomorphisms with non-zero Lyapunov exponents that are Bernoulli, exhibit polynomial decay of correlations, and satisfy statistical properties like the CLT and large deviations.

Contribution

It introduces a method to construct surface diffeomorphisms with specified statistical and dynamical properties, including polynomial decay of correlations and Bernoulli behavior.

Findings

Existence of area-preserving diffeomorphisms with polynomial decay of correlations.

Diffeomorphisms satisfy the Central Limit Theorem and Large Deviation Property.

Presence of a unique hyperbolic Bernoulli measure with exponential decay of correlations.

Abstract

We show that any surface admits an area preserving $C^{1+\beta}$ diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations. In addition, we show that this diffeomorphism satisfies the Central Limit Theorem and has the Large Deviation Property. Finally, we show that the diffeomorphism we constructed possesses a unique hyperbolic Bernoulli measure of maximal entropy with respect to which it has exponential decay of correlations.