Open Sets of Exponentially Mixing Anosov Flows

TL;DR

This paper proves that certain Anosov flows with non-jointly integrable stable and unstable bundles exhibit exponential mixing, and identifies open sets of such flows, including specific suspension semiflows.

Contribution

It establishes exponential mixing for Anosov flows with   stable bundles and non-joint integrability, extending to nearby flows and suspension semiflows.

Findings

Anosov flows with   stable bundle mix exponentially if stable and unstable bundles are not jointly integrable.

Open sets of exponentially mixing Anosov flows exist.

Suspension semiflows with non-cohomologous return times mix exponentially.

Abstract

We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniformly-expanding suspension semiflows (in any dimension) mix exponentially when the return time in not cohomologous to a piecewise constant.