2021/07/25 by Tahar, Guillaume · 1 citation
#37E35 #57M50 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2107.11745
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions of its hyperbolic closed geodesics form a dense subset of \mathbbS1. We also prove that a dilation surface satisfies the latter property if and only if its directional flow is Morse-Smale in an open dense subset of \mathbbS1.