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Bellis strong stable sets on infinite hyperbolic surfaces

Sergi Burniol Clotet, Françoise Dal'Bo, Sergio Herrero Vila

Abstract

We provide a corrected proof of a theorem of A. Bellis on strong stable sets in the unit tangent bundle of certain hyperbolic surfaces. The theorem states that, for vectors whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set in the dynamical sense does not coincide with the associated horocyclic orbit. The proof is based on Bellis' idea of constructing geodesic rays that wind around infinitely many closed geodesics.

Bellis strong stable sets on infinite hyperbolic surfaces

Abstract

We provide a corrected proof of a theorem of A. Bellis on strong stable sets in the unit tangent bundle of certain hyperbolic surfaces. The theorem states that, for vectors whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set in the dynamical sense does not coincide with the associated horocyclic orbit. The proof is based on Bellis' idea of constructing geodesic rays that wind around infinitely many closed geodesics.

Paper Structure

This paper contains 8 sections, 8 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Winding around a closed geodesic
  4. Proof of Main Theorem
  5. Definition of $v_n$ and $w_\alpha$
  6. Construction of suitable $\alpha=(\alpha_n)_{n\in\mathbb{N} }$
  7. End of the Proof of the Main Theorem
  8. Equality between $W_{d_1}^{ss}(u)$ and $W_{d_2}^{ss}(u)$

Key Result

Proposition 1

Let $\tilde{u}\in T^{1}\mathbb{H}^{2}$ and $\gamma$ a hyperbolic isometry such that the geodesic ray $\tilde{u}(\mathbb{R}^+)$ meets the axis $(\gamma^-,\gamma^+)$, then $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Winding around a closed geodesic in the universal covering, where $t_0$ the time such that $\tilde{u}(t_0)\in (\gamma^-,\gamma^+)$.
  • Figure 2: $g_{r_n}\beta_n^{-1}\tilde{v}_n\in h_\mathbb{R}\tilde{u}$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • proof
  • Definition 5
  • Definition 6: winding time
  • Proposition 1: bound of the winding time
  • proof
  • ...and 15 more