Decay of correlations for certain isometric extensions of Anosov flows

TL;DR

This paper proves exponential decay of correlations for certain isometric extensions of transitive Anosov flows, using non-integrability estimates and transfer operators, under specific smoothness conditions.

Contribution

It introduces a method to establish decay of correlations for isometric extensions of Anosov flows by linking accessibility to non-integrability and transfer operator analysis.

Findings

Exponential decay of correlations established for these extensions.

Method applies to all orders of correlations.

Results depend on joint $C^1$ smoothness of foliations.

Abstract

We establish exponential decay of correlations of all orders for locally $G$-accessible isometric extensions of transitive Anosov flows, under the assumption that the strong stable and strong unstable foliations of the base Anosov flow are jointly $C^1$. This is accomplished by translating accessibility properties of the extension into local non-integrability estimates measured by Dolgopyat's infinitesimal transitivity group, from which we obtain contraction properties for a class of 'twisted' symbolic transfer operators.