Bernoulli equilibrium states for surface diffeomorphisms

TL;DR

This paper proves that for certain surface diffeomorphisms with positive entropy, the system's measure-theoretic structure is equivalent to a Bernoulli process combined with a finite rotation, revealing a deep stochastic property.

Contribution

It establishes a Bernoulli isomorphism for equilibrium states of surface diffeomorphisms with positive entropy, linking dynamical systems to Bernoulli processes.

Findings

Surface diffeomorphisms with positive entropy are Bernoulli up to a finite rotation.

Equilibrium measures of Holder continuous potentials exhibit Bernoulli properties.

The result applies to $C^{1+eta}$ surface diffeomorphisms with positive entropy.

Abstract

Suppose f is a $C^{1+\alpha}$ surface diffeomorphism, and m is an equilibrium measure of a Holder continuous potential. We show that if m has positive metric entropy, then f is measure theoretically isomorphic to the product of a Bernoulli scheme and a finite rotation.