  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 $\mathbb{S}^{1}$. 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 $\mathbb{S}^{1}$.
