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

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.