Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space

Yannick Guedes Bonthonneau

doi:10.33044/revuma.v61n1a03

Abstract

Given an Anosov vector field X-0, all sufficiently close vector fields are also of Anosov type. In this note, we check that the anisotropic spaces described by Faure and Sjostrand and by Dyatlov and Zworski can be chosen adapted to any smooth vector field sufficiently close to X-0 in C-1 norm

Highlights

M will denote a compact manifold of dimension n, and X0 a smooth Anosov vector field on M
Starting with [2], several authors have built some anisotropic spaces of distributions to study the spectral properties of hyperbolic dynamics, of which Anosov flows are a prime example
This enables the study of so-called Ruelle–Pollicott resonances, originally defined using the techniques of Markov partitions [14, 13]. They appeared as the poles of some zeta functions, popularized by Smale [15]

Summary

Introduction

M will denote a compact manifold of dimension n, and X0 a smooth Anosov vector field on M. Starting with [2], several authors have built some anisotropic spaces of distributions to study the spectral properties of hyperbolic dynamics, of which Anosov flows are a prime example. This enables the study of so-called Ruelle–Pollicott resonances, originally defined using the (quite different) techniques of Markov partitions [14, 13]. BONTHONNEAU (here Op denotes a classical quantization, and G ∈ Slog(T ∗M )) Using these technique led to new developments. Chaubet for the discussion that led to Lemma 4.3

Microlocal proof of the main theorem

Building the weight function

Perturbation of resonances