On the Bernoulli property for certain partially hyperbolic diffeomorphisms

TL;DR

This paper proves that for a specific class of volume-preserving partially hyperbolic diffeomorphisms on the 3-torus, the Kolmogorov and Bernoulli properties are equivalent, establishing a strong ergodic classification.

Contribution

It demonstrates the equivalence of Kolmogorov and Bernoulli properties for derived from Anosov diffeomorphisms on the 3-torus and introduces a method analyzing conditional measures along central foliations.

Findings

Kolmogorov and Bernoulli properties are equivalent for the studied class.

Existence of an almost everywhere conjugacy to the linearization.

Results extend to higher dimensions with one-dimensional central bundle.

Abstract

We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization. Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.