Horizon saddle connections and Morse-Smale dynamics of dilation surfaces

Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties of positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may feature trajectories accumulating on a limit cycle. These limit cycles are closed geodesics called hyperbolic because they induce a nontrivial homothety. It has been conjectured that a dilation surface without any hyperbolic closed geodesic is in fact a translation surface. Assuming existence of a horizon saddle connection in a dilation surface, we prove that the directions of its hyperbolic closed geodesics form a dense subset of $\mathbb{S}^{1}$. 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 $\mathbb{S}^{1}$.