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The geodesic flow on a hyperbolic surface with a cusp is not expansive

Sergi Burniol Clotet, Françoise Dal'Bo

Abstract

We prove that the geodesic flow on any hyperbolic surface $S$ with at least one cusp is not expansive. The proof is based on the study of strong-stable sets.

The geodesic flow on a hyperbolic surface with a cusp is not expansive

Abstract

We prove that the geodesic flow on any hyperbolic surface with at least one cusp is not expansive. The proof is based on the study of strong-stable sets.

Paper Structure

This paper contains 15 sections, 19 theorems, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Horocycles and boundary at infinity
  4. Distance on the unit tangent bundle
  5. Fuchsian groups
  6. Cusps and parabolic isometries
  7. Some technique lemmas
  8. Winding around a closed horocycle
  9. Proof of Theorem \ref{['thm2']}
  10. Cusp-recurrent vectors
  11. Construction of $w_\alpha$ in $\overline{H^{ss} u }$
  12. Construction of $w^\alpha$ in $W^{ss} u$
  13. Construction of $w_\alpha$ in $W^{ss} u \setminus H^{ss} u$
  14. Abundance of cusp-recurrent vectors
  15. Non expansive geodesic flow

Key Result

Theorem 1

Let $S$ be a nonelementary hyperbolic surface with at least one cusp. Then the geodesic flow on $T^1 S$ is not expansive.

Figures (4)

  • Figure 1: Winding of $\tilde{u}$ around the pair $(\tilde{H}, p)$.
  • Figure 2: Radii and centers of the half-circles associated to $\tilde{u}$ and $\tilde{v}$.
  • Figure 3: Position of the sequence of horocycles $\tilde{H}_n$.
  • Figure 4: Region $R_n$ and ray of $\beta_n^{-1} \tilde{v}_n$.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 2: bound of the winding time
  • ...and 28 more