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$C^0$-Contact Anosov flows

Cheikh Khoule, Matheus Manso, Ameth Ndiaye, Khadim War

Abstract

We prove that smooth reparametrizations of the geodesic flow on a manifold of constant negative curvature are contact Anosov flows. In particular we give a new class of exponentially mixing Anosov flows. Moreover, this introduces the notion of $C^0$-contact and we prove that the classical Gray stability theorem that is known in the smooth case fails in this setting.

$C^0$-Contact Anosov flows

Abstract

We prove that smooth reparametrizations of the geodesic flow on a manifold of constant negative curvature are contact Anosov flows. In particular we give a new class of exponentially mixing Anosov flows. Moreover, this introduces the notion of -contact and we prove that the classical Gray stability theorem that is known in the smooth case fails in this setting.

Paper Structure

This paper contains 25 sections, 36 theorems, 240 equations, 6 figures.

Table of Contents

  1. Introduction and results
  2. Sketch:
  3. Structure:
  4. Exponentially mixing
  5. Deformation and Gray stability
  6. Geometric Background
  7. Ideal boundary $\partial\widetilde{M}$ and Busemann function
  8. Divergence property:
  9. Ideal boundary $\partial \widetilde{M}$:
  10. Notations:
  11. Busemann function:
  12. Connecting points and the strips:
  13. Stable manifolds:
  14. Axis of isometry element:
  15. Algebraic Anosov flows and reparametrizations
  16. ...and 10 more sections

Key Result

Theorem A

Let $(M,g)$ be a closed Riemannian manifold of constant sectional curvatures $-1.$ Let $Z$ be the geodesic vector field and $\alpha$ be the canonical $1$-form defined in eq:al. If $\psi: SM\to(0,\infty)$ is a $C^\infty$ positive function then the vector field $Z_\psi$ generates a contact Anosov flow where $\overline\psi:=\int_{SM}\psi dm$ and $m$ is the Liouville measure.

Figures (6)

  • Figure 1: Geometric interpretation of $b_p(q,\xi)$ and $\beta_p(\xi,\eta)$.
  • Figure 2: Dynamical interpretation of Cross ratio
  • Figure 3: Cross ratio and reparametrization
  • Figure 4: Cross ratio
  • Figure 5: Exterior derivative
  • ...and 1 more figures

Theorems & Definitions (86)

  • Theorem A
  • Theorem B
  • Theorem 1.3
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 76 more