R\'ecurrence ou non minimalit\'e des adh\'erences des d'orbites irr\'eguli\'eres du flot horocyclique de finesse infinie

TL;DR

This paper investigates the recurrence and minimality properties of irregular orbits of the horocyclic flow on geometrically infinite hyperbolic surfaces, extending understanding beyond the well-known finite case.

Contribution

It proves that irregular orbits are either recurrent or have non-minimal closures, advancing the classification of minimal sets for the horocyclic flow on infinite surfaces.

Findings

Irregular orbits are recurrent or have non-minimal closures.

This result helps complete the description of minimal sets for the flow.

The study extends known results from finite to infinite hyperbolic surfaces.

Abstract

The topological dynamics of the horocyclic flow $h_{\mathbb{R}}$ on the unit tangent bundle of a geometrically finite hyperbolic surface is well known. In particular, on such a surface, the flow $h_{\mathbb{R}}$ 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 non-closed and non-dense orbits, called irregular orbits, complicates the description of minimal sets. In this text, we will show that such an orbit is recurrent, or its closure is non-$h_{\mathbb{R}}$ minimal. This would allow us to almost complete the description of $h_{\mathbb{R}}$-minimal sets.