Stochastic Stability of Pollicott-Ruelle Resonances Open Access

Abstract

This dissertation studies the stochastic stability of Pollicott–Ruelle resonances for hyperbolic dynamical systems. The main results show that adding a small diffusive (elliptic) perturbation produces an operator with a discrete spectrum whose eigenvalues converge, locally uniformly in the complex plane, to the Pollicott–Ruelle resonances of the unperturbed dynamics. We establish this convergence in two settings: open hyperbolic systems (via an extension to a compact ambient manifold and microlocal complex absorbing operators) and Axiom A flows satisfying a strong transversality condition. The proofs rely on anisotropic Sobolev spaces and microlocal/semiclassical propagation estimates.

Table of Contents

Table of Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Dynamical Preliminaries 8

2.1 Definitions of open systems and Axiom A systems . . . . . . . . . . . . . . . 8

2.2 Dynamics on the Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 14

3 Microlocal Preliminaries. 17

3.1 Microlocal Analysis on Euclidean spaces . . . . . . . . . . . . . . . . . . . . 17

3.2 Microlocal Analysis on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Microlocal Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Stochastic Stability for Open Hyperbolic Systems 25

4.1 Anisotropic Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Complex Absorbing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 An Intermediate Elliptic Estimate . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Main Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Stochastic Stability for Axiom A Systems 46

5.1 Anisotropic Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Bibliography 57

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Physics, Quantum Mathematics
Keyword	Hyperbolic Dynamics Partial Differential Equations Microlocal Analysis Spectral Theory Pollicott-Ruelle Resonances
Committee Chair / Thesis Advisor	David Borthwick, Emory University
Committee Members	Levon Nurbekyan, Emory University Yiran Wang, Emory University

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