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Geodesics rays diverging on average on a pair of pants

Lo Cheikh, Vila Sergio

TL;DR

The paper studies endpoints of geodesic rays that diverge on average on a one-cusp hyperbolic pair of pants by developing an explicit coding for the diverging-on-average set $\Lambda_{\infty}$ associated with a Schottky group $\Gamma=\langle h,p\rangle$. It proves a precise criterion for membership in $\Lambda_{\infty}$ based on excursions and cusp windings, and establishes the fractal size of this set by showing $dim_H(\Lambda_{\infty})=\tfrac{1}{2}$ via matching upper and lower bounds. The approach combines geometric coding, shadowing arguments, and Frostman-type measures to connect dynamical behavior on the hyperbolic surface with the Hausdorff dimension of a limit-set subset. This work extends prior results on divergence on average by providing a self-contained, explicit calculation of the critical dimension.

Abstract

The aim of this paper is to characterize in terms of coding a set of limit points considered in a paper of F. Riquelme and A. Velozo corresponding to geodesic rays which spend less time in any compact region of a pair of pants with one cusp. Moreover in this particular context we reprove that the Hausdorff dimension of this set is equal 1/2 by explicit calculus.

Geodesics rays diverging on average on a pair of pants

TL;DR

The paper studies endpoints of geodesic rays that diverge on average on a one-cusp hyperbolic pair of pants by developing an explicit coding for the diverging-on-average set associated with a Schottky group . It proves a precise criterion for membership in based on excursions and cusp windings, and establishes the fractal size of this set by showing via matching upper and lower bounds. The approach combines geometric coding, shadowing arguments, and Frostman-type measures to connect dynamical behavior on the hyperbolic surface with the Hausdorff dimension of a limit-set subset. This work extends prior results on divergence on average by providing a self-contained, explicit calculation of the critical dimension.

Abstract

The aim of this paper is to characterize in terms of coding a set of limit points considered in a paper of F. Riquelme and A. Velozo corresponding to geodesic rays which spend less time in any compact region of a pair of pants with one cusp. Moreover in this particular context we reprove that the Hausdorff dimension of this set is equal 1/2 by explicit calculus.

Paper Structure

This paper contains 11 sections, 15 theorems, 84 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background and statements of results
  3. The diverging on average set
  4. Behavior of geodesic rays in the surface
  5. Cusp crossing time
  6. Geometric encoding of the limit set
  7. Proof of theorem \ref{['average']}
  8. Examples
  9. Hausdorff dimension of the diverging on average set
  10. The upper bound
  11. The lower bound

Key Result

Theorem 1

Let $\Gamma$ be a Schottky group generated by a hyperbolic isometry $h$ and a parabolic isometry $p$. Then $\xi\in\Lambda_{\infty}$ if only if for any positive integer $N$ there exists a coding in terms $\omega_{i}$ and $p^{r_{i}}$ such that where $d$ is the hyperbolic distance on $\mathbb{D}$ and for every $i\in[1,q]\;\;\hbox{and}\;\;k\in [1,s_{i}]$

Figures (10)

  • Figure 1: Dirichlet (Ford) domain of $\Gamma$ centered at $0$.
  • Figure 2: The surface $\Sigma_0$.
  • Figure 3: A lift of an $(n,k)$-excursion of a geodesic ray on $\mathbb{H}$
  • Figure 4: The highest horocycle that $\pi([i,\xi))$ can meet is $\pi(\mathcal{HD}_{\infty}(N))$).
  • Figure 5: The geodesic segment $\sigma|_{[0,T]}$ winds in a cusp and returns in the corresponding compact
  • ...and 5 more figures

Theorems & Definitions (33)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2: Corollary 2.2 in Vo
  • Lemma 1
  • proof
  • ...and 23 more