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Recurrent and irregular orbits of the horocyclic flow on the unit tangent bundle of the untwisted flute

Amadou Sy

TL;DR

The paper investigates the horocycle flow on the unit tangent bundle of untwisted flute surfaces, focusing on when the horocycle orbit closure intersects the geodesic orbit of a vector $u$. By leveraging horocyclic convergence criteria, limit-set stratification, and the geometry of geometrically infinite Fuchsian groups, the authors prove that for a quasi-minimizing half-geodesic $u(\mathbb{R}^{+})$ avoiding all closed geodesics and with $h_{\mathbb{R}}u$ neither closed nor dense, the set $T_{u}=\{t\in\mathbb{R}: g_t u\in \overline{h_{\mathbb{R}}u}\}$ is trivial, i.e., $T_{u}=\{0\}$. They construct an explicit untwisted flute from a nested-Schottky group to realize an infinite quasi-minimizing ray with infinite asymptotic fineness and non-dense horocycle dynamics, yet still satisfy $T_{u}=\{0\}$. The work also proves structural properties of $T_{u}$ as a nonempty closed semigroup and provides a concrete infinite-example showing $T_{u}$ can be infinite in other setups. These results illuminate how horocycle non-minimality behaves in infinite-type hyperbolic surfaces and respond to questions about closures of horocycle orbits on flutes. The findings have implications for understanding the interaction between horocycle and geodesic dynamics on geometrically infinite surfaces and for Bellis’ questions on flute surfaces.

Abstract

The aim of this article is to show that if there exists $u \in Ω_{h} \subset T^{1}S$ an infinite quasi-minimizing ray which do not intersect any closed geodesic on the surface $S$ (untwisted flute), then $T_{u}=\{ t \in \mathbb{R} \; ; \; g_{t}u \in \overline{h_{\mathbb{R}}u} \}=\{0\}$.

Recurrent and irregular orbits of the horocyclic flow on the unit tangent bundle of the untwisted flute

TL;DR

The paper investigates the horocycle flow on the unit tangent bundle of untwisted flute surfaces, focusing on when the horocycle orbit closure intersects the geodesic orbit of a vector . By leveraging horocyclic convergence criteria, limit-set stratification, and the geometry of geometrically infinite Fuchsian groups, the authors prove that for a quasi-minimizing half-geodesic avoiding all closed geodesics and with neither closed nor dense, the set is trivial, i.e., . They construct an explicit untwisted flute from a nested-Schottky group to realize an infinite quasi-minimizing ray with infinite asymptotic fineness and non-dense horocycle dynamics, yet still satisfy . The work also proves structural properties of as a nonempty closed semigroup and provides a concrete infinite-example showing can be infinite in other setups. These results illuminate how horocycle non-minimality behaves in infinite-type hyperbolic surfaces and respond to questions about closures of horocycle orbits on flutes. The findings have implications for understanding the interaction between horocycle and geodesic dynamics on geometrically infinite surfaces and for Bellis’ questions on flute surfaces.

Abstract

The aim of this article is to show that if there exists an infinite quasi-minimizing ray which do not intersect any closed geodesic on the surface (untwisted flute), then .
Paper Structure (15 sections, 23 theorems, 41 equations, 3 figures)

This paper contains 15 sections, 23 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence in $T^{1}S$, Asymptotic Thinness, and Limit Points
  4. Characterization of the Elements of $T_{u}$
  5. Asymptotic Thinness
  6. Nature of the points of the limit set
  7. Correspondence between limit points and the topological nature of orbits.
  8. Proof of Theorem \ref{['pa']}
  9. Proof of Theorem \ref{['TAHA']}
  10. Presentation of the Group and Construction of the Untwisted Flute
  11. Asymptotic fineness of an infinite quasi-minimizing ray
  12. No-closed nor-dense Orbit
  13. Let us show that $T_u = \{ t \in \mathbb{R} : g_t u \in \overline{h_{\mathbb{R}} u} \} = \{0\}$.
  14. properties of $T_{u}$
  15. Example in the case that $T_{u}$ is infinite

Key Result

Theorem 1.1

Let $S$ be an untwiste flute and $u \in \Omega_{h} \subset T^{1}S$ such that the horocyclic orbit $h_{\mathbb{R}}u$ is neither closed nor dense in the non-wandering set of the horocyclic flow. If $u(\mathbb{R^{+}})$ does not intersect any closed geodesic on the surface $S$, then

Figures (3)

  • Figure 1: Hyperbolic polygon
  • Figure 2: Fundamental domain.
  • Figure 3: untwisted flute (Surface)

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2: Untwisted and Twisted Flutes
  • Proposition 1
  • Corollary 3.1
  • Corollary 3.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • ...and 30 more