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Mathematics > Dynamical Systems
Title: Horizon saddle connections and Morse-Smale dynamics of dilation surfaces
(Submitted on 25 Jul 2021 (v1), last revised 9 Feb 2023 (this version, v2))
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 $\mathbb{S}^{1}$.
Submission history
From: Guillaume Tahar [view email][v1] Sun, 25 Jul 2021 07:34:17 GMT (59kb,D)
[v2] Thu, 9 Feb 2023 07:51:36 GMT (223kb,D)
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