Isometric extensions of Anosov flows via microlocal analysis

TL;DR

This paper applies microlocal analysis to study ergodicity and mixing in isometric extensions of Anosov flows, providing new insights into the spectral properties and ergodic behavior of these dynamical systems.

Contribution

It introduces a microlocal framework to analyze ergodicity of isometric extensions of Anosov flows, extending classical results with novel analytical techniques.

Findings

Proves ergodicity of the frame flow under nearly pinched curvature conditions.

Connects microlocal analysis with dynamical properties of Anosov flows.

Provides spectral analysis tools for studying ergodic extensions.

Abstract

We revisit the classical framework developed by Brin, Pesin and others to study ergodicity and mixing properties of isometric extensions of volume-preserving Anosov flows, using the microlocal framework developed in the theory of Pollicott-Ruelle resonances. The approach developed in the present note is reinvested in the companion paper [arXiv:2111.14811] in order to show ergodicity of the frame flow on negatively-curved Riemannian manifolds under nearly $1/4$-pinched curvature assumption (resp. nearly $1/2$-pinched) in dimension $4$ and $4\ell+2, \ell > 0$ (resp. dimension $4\ell, \ell > 0$).