Ergodic properties of equilibrium measures for smooth three dimensional flows

TL;DR

This paper investigates the ergodic properties of equilibrium measures for smooth three-dimensional flows, establishing conditions under which such flows are Bernoulli or decomposable into Bernoulli and rotational components, with applications to Reeb flows.

Contribution

It characterizes the ergodic measures of maximal entropy for smooth three-dimensional flows, showing they are either Bernoulli or a product of Bernoulli and rotational flows, extending understanding of flow dynamics.

Findings

Flows are Bernoulli or product of Bernoulli and rotational flows.

Application to Reeb flows demonstrates broader relevance.

Provides classification of ergodic measures for these flows.

Abstract

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $\mu$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.