Horizon saddle connections and Morse-Smale dynamics of dilation surfaces

TL;DR

This paper explores the dynamics of dilation surfaces, showing that if such a surface has a horizon saddle connection, then hyperbolic geodesics are densely distributed in directions, and relates this to Morse-Smale flows.

Contribution

It establishes a connection between horizon saddle connections, the density of hyperbolic geodesic directions, and Morse-Smale dynamics on dilation surfaces.

Findings

Hyperbolic geodesic directions are dense if a horizon saddle connection exists.

Dilation surfaces with dense hyperbolic geodesic directions have Morse-Smale flows in a dense subset.

The paper characterizes when dilation surfaces exhibit Morse-Smale dynamics based on their geodesic structure.

Abstract

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