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Counting Pollicott--Ruelle resonances for Axiom A flows

Long Jin, Zhongkai Tao

Abstract

In this paper, we count the number of Pollicott--Ruelle resonances for open hyperbolic systems and Axiom A flows. In particular, we prove polynomial upper bounds and sublinear lower bounds on the number of resonances with modulus less than $r$ in strips for open hyperbolic systems and Axiom A flows with a transversality condition.

Counting Pollicott--Ruelle resonances for Axiom A flows

Abstract

In this paper, we count the number of Pollicott--Ruelle resonances for open hyperbolic systems and Axiom A flows. In particular, we prove polynomial upper bounds and sublinear lower bounds on the number of resonances with modulus less than in strips for open hyperbolic systems and Axiom A flows with a transversality condition.
Paper Structure (30 sections, 32 theorems, 208 equations)

This paper contains 30 sections, 32 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Previous works and the method of the proof
  3. Organization of the paper
  4. Setup and Preliminaries
  5. Dynamics of open hyperbolic systems
  6. Dynamics on the cotangent bundle
  7. Preliminaries on semiclassical and microlocal analysis
  8. Spectral theory of open hyperbolic systems
  9. Upper bound on the number of resonances
  10. Modification of the complex absorbing operator
  11. Resolvent estimates
  12. Proof of the upper bound
  13. Lower bound on the number of resonances
  14. Support conditions
  15. A wavefront set estimate for the modified resolvent
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $n=\dim M$. Suppose $\varphi^t=e^{tX}$ is an Axiom A flow on $M$ satisfying the strong transversality condition, then for any fixed $\beta>0$, there exists $C>0$ such that for any $E>0$, If in addition, $\varphi^t$ has at least one closed orbit, then for any $\delta\in(0,1)$ there exist $\beta>0$ and $C>0$ depending on $\delta$ such that

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 48 more