Mixing implies exponential mixing among codimension one hyperbolic attractors and Anosov flows

TL;DR

This paper proves that certain hyperbolic attractors and Anosov flows on manifolds of dimension three or higher exhibit exponential mixing, with the results applying broadly to Axiom A vector fields and confirming a longstanding conjecture.

Contribution

It establishes exponential mixing for hyperbolic attractors with non-integrable foliations and confirms the Bowen-Ruelle conjecture for codimension one Anosov flows.

Findings

Exponential mixing occurs under joint non-integrability conditions.

A dense and open set of Axiom A vector fields exhibit exponential mixing.

Exponential mixing is proven for codimension one Anosov flows, confirming the Bowen-Ruelle conjecture.

Abstract

On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the flow mixes with respect to the unique physical measure, then the flow mixes exponentially, proving the Bowen-Ruelle conjecture in this setting.