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

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.