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On the closure of irregular orbits of the horocyclic flow on infinite finness

Amadou Sy, Masseye Gaye

TL;DR

This work addresses minimal sets for the horocyclic flow on the unit tangent bundle of geometrically infinite hyperbolic surfaces. It develops a constructive proof by building a one-parameter family of surfaces $\Sigma_{\delta}$ via a carefully designed Fuchsian group $\Gamma$, yielding a geometrically infinite surface with an irregular horocycle orbit whose closure is not $h_{\mathbb{R}}$-minimal. The authors prove the existence of an irregular orbit $h_{\mathbb{R}}(u_0)$ with asymptotic fineness $Inj(u_0(\mathbb{R}_{+}))=+\infty$ and show that its associated set of times $T_{u_0}$ contains a nontrivial, Fibonacci-like sequence $(t_n)$ with $t_0=\ln(\delta^2)$ and $t_1=2\ln(\delta(\delta+1))$, establishing new non-minimality phenomena for infinite-type hyperbolic surfaces. The results deepen understanding of horocycle dynamics in the infinite-end regime and provide concrete mechanisms linking limit-set structure, horocyclic convergence, and minimal set theory.

Abstract

The topological dynamics of the horocyclic flow h_R on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular on such a surface the flow h_R is minimal or the minimal sets are the periodic orbits. When the surface is geometrically infinite, the situation is more complex and the presence of possible irregular orbits makes the description of minimal sets complicated. In this text, we construct a family of infinite hyperbolic surfaces for which the horocyclic flow defined on the unit tangent bundle is not minimal.

On the closure of irregular orbits of the horocyclic flow on infinite finness

TL;DR

This work addresses minimal sets for the horocyclic flow on the unit tangent bundle of geometrically infinite hyperbolic surfaces. It develops a constructive proof by building a one-parameter family of surfaces via a carefully designed Fuchsian group , yielding a geometrically infinite surface with an irregular horocycle orbit whose closure is not -minimal. The authors prove the existence of an irregular orbit with asymptotic fineness and show that its associated set of times contains a nontrivial, Fibonacci-like sequence with and , establishing new non-minimality phenomena for infinite-type hyperbolic surfaces. The results deepen understanding of horocycle dynamics in the infinite-end regime and provide concrete mechanisms linking limit-set structure, horocyclic convergence, and minimal set theory.

Abstract

The topological dynamics of the horocyclic flow h_R on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular on such a surface the flow h_R is minimal or the minimal sets are the periodic orbits. When the surface is geometrically infinite, the situation is more complex and the presence of possible irregular orbits makes the description of minimal sets complicated. In this text, we construct a family of infinite hyperbolic surfaces for which the horocyclic flow defined on the unit tangent bundle is not minimal.
Paper Structure (9 sections, 8 theorems, 28 equations)

This paper contains 9 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nature of the points of the limit set
  4. Topological nature of the orbits of the horocyclic flow
  5. Horocyclic convergence
  6. Proof of the theorem \ref{['pa']}
  7. Construction of the surface $\Sigma_{\delta}$
  8. Existence of the irregular orbit $h_{\mathbb R}(u_0)$ and its properties
  9. Asymptotic fineness of $u_0(\mathbb R_{+})$.

Key Result

Theorem 1.1

There exists a family of geometrically infinite hyperbolic surfaces $\Sigma _ {\delta}$ for which there exists $u_0\in \Omega_h$ satisfying : 1. $h _ {\mathbb R} (u)$ is a irregular orbit with asymptotic fineness $Inj(u(\mathbb R _ {+})) = +\infty$ and the closure $\overline{h_{\mathbb R}(u_0)}$ is

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6