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FBI Transform in Gevrey Classes and Anosov Flows

Yannick Guedes Bonthonneau, Malo Jézéquel

Abstract

An analytic FBI transform is built on compact manifolds without boundary, that satisfies all the expected properties. It enables the study of microlocal analytic and Gevrey regularity on such manifolds. This tool is then used to study the Ruelle spectrum of Anosov flows with Gevrey coefficients. In particular, finite order for the associated dynamical determinant is proved.

FBI Transform in Gevrey Classes and Anosov Flows

Abstract

An analytic FBI transform is built on compact manifolds without boundary, that satisfies all the expected properties. It enables the study of microlocal analytic and Gevrey regularity on such manifolds. This tool is then used to study the Ruelle spectrum of Anosov flows with Gevrey coefficients. In particular, finite order for the associated dynamical determinant is proved.

Paper Structure

This paper contains 47 sections, 115 theorems, 869 equations, 3 figures.

Table of Contents

  1. Gevrey microlocal analysis on manifolds
  2. Gevrey spaces of functions and symbols
  3. Gevrey functions and ultradistributions on manifolds
  4. First definitions
  5. The particular case of real-analytic functions
  6. Almost analytic extensions
  7. Gevrey and analytic symbols
  8. Definitions
  9. Hörmander's problem on Grauert tube and analytic approximations
  10. Gevrey oscillatory integrals
  11. Non-stationary phase
  12. Gevrey Stationary Phase
  13. Further tricks
  14. Gevrey pseudo-differential operators
  15. Gevrey and analytic pseudo-differential operators on compact manifolds
  16. ...and 32 more sections

Key Result

Theorem 1

Let $s \in \left[1, + \infty\right[$. Let $M$ be an $n$-dimensional compact $s$-Gevrey manifold endowed with a $s$-Gevrey Anosov flow $\left( \phi_t \right)_{t \in \mathbb R}$ generated by a vector field $X$. Then there is a constant $C>0$ such that for every $z\in \mathbb C$, In particular, $\zeta_X$ has finite order less than $ns$.

Figures (3)

  • Figure 1: Sketch of the contour deformation.
  • Figure 2: The complex neighbourhoods involved in the extension of the kernel for $s=1$.
  • Figure 3: Region near the diagonal in the complex where $K_{TPS}$ can be controlled. Here, $\tilde{h} = h/\langle|\alpha|\rangle$.

Theorems & Definitions (273)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.1
  • Lemma 1.2
  • ...and 263 more