Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Critical axis of Ruelle resonances for Anosov flow with a potential

Tristan Humbert

Abstract

We combine methods from microlocal analysis and dimension theory to study resonances with largest real part for an Anosov flow with smooth real valued potential. We show that the resonant states are closely related to special systems of measures supported on the stable manifolds introduced by Climenhaga. As a result, we relate the presence of the resonances on the critical axis to mixing properties of the flow with respect to certain equilibrium measures and show that these equilibrium measures can be reconstructed from the spectral theory of the Anosov flow.

Critical axis of Ruelle resonances for Anosov flow with a potential

Abstract

We combine methods from microlocal analysis and dimension theory to study resonances with largest real part for an Anosov flow with smooth real valued potential. We show that the resonant states are closely related to special systems of measures supported on the stable manifolds introduced by Climenhaga. As a result, we relate the presence of the resonances on the critical axis to mixing properties of the flow with respect to certain equilibrium measures and show that these equilibrium measures can be reconstructed from the spectral theory of the Anosov flow.
Paper Structure (20 sections, 24 theorems, 171 equations, 1 figure)

This paper contains 20 sections, 24 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Leading resonant state
  3. Equilibrium states.
  4. Ruelle zeta function.
  5. Regularity of the pressure.
  6. Complex potential
  7. Preliminaries
  8. Anosov flow
  9. Thermodynamical formalism
  10. Anisotropic spaces
  11. Equilibrium states from dimension theory
  12. Resolvent acting on functions
  13. Construction of the co-resonant state
  14. Critical axis
  15. Resonances on the critical axis
  16. ...and 5 more sections

Key Result

Theorem 1

Under Assumption assumption, the critical axes for the action on $0$-forms and $d_s$-forms are given by: Moreover, $P(V+J^u)$$($ resp. $P(V))$ is a resonance called, the first resoance for the action on $0$-forms $($ resp. $d_s$-forms). There is $\delta>0$ such that for any $k\neq d_s$, we have $\mathcal{C}_k\subset \{\lambda \mid \mathrm{Re}(\lambda)\leq P(V)-\delta\}$, i.e all other critical ax

Figures (1)

  • Figure 1: Critical axes for different values of $k$. According to Theorem \ref{['mainTheo']}, the resonances in purple cannot exist if the flow is weakly mixing with respect to $\mu_V$ and the resonances in blue cannot exist if the flow is weakly mixing with respect to $\mu_{V+J^u}$. The position of the critical axes for intermediate values of $k$ should be linked to the pressure on the span of largest Lyapunov exponents.

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Theorem : Parry-Pollicott
  • Corollary 1: Critical axis of the zeta function
  • Corollary 2: Smoothness of the topological pressure
  • Corollary 3: Complex potential
  • Definition 2.1: Anosov
  • Theorem 3: Faure Sjöstrand
  • Remark 1
  • Remark 2
  • ...and 36 more