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Isometric extensions of Anosov flows via microlocal analysis

Thibault Lefeuvre

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 $4\ell, \ell > 0$).

Isometric extensions of Anosov flows via microlocal analysis

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 -pinched curvature assumption (resp. nearly -pinched) in dimension and (resp. dimension ).
Paper Structure (10 sections, 21 theorems, 69 equations)

This paper contains 10 sections, 21 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isometric extensions of Anosov flows
  4. Invariant sections on vector bundles
  5. Isometric actions on closed manifolds
  6. Proof of the Theorem
  7. Ergodicity
  8. Mixing
  9. The case of principal bundles
  10. On volume-preserving Anosov flows

Key Result

Theorem 1

Under the above assumptions, the followings holds:

Theorems & Definitions (47)

  • Theorem
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 37 more