Horocycle orbits on surfaces of negative curvature

The horocycle flow on a surface of negative curvature is closely related to the geodesic
flow, which also has hyperbolicity properties. On the context of constant negative
curvature, invariant probability measures for the horocycle flow have been precisely
described already, but very little is known about the closure of the orbits when the
surface has infinite volume, and particularly when it is of infinite type.
Recently, Matsumoto studied some particular class of surfaces, that appear naturally
on the study of some laminations by hyperbolic surfaces, and proved that on those
surfaces, the horocycle flow does not admit minimal sets. In that context he also proved
that almost minimizing geodesic rays have accumulation points of the horocycle orbit
of its points. Later, Bellis proved on a more general context that almost minimizing
geodesic rays have infinite accumulation points of the horocycle orbits of its points.
The aim of this talk will be presenting the general objects and ideas involved on these
results, which have been proved for surfaces of constant negative curvature, and showing
the difficulties in generalizing them to the context of variable negative curvature.