2019/08/26 by Guillaume Tahar, Tahar, Guillaume
Mathematics · #53A15 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1908.09595
openalex publication_date 2019/08/26 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
Dilation surfaces are generalizations of translation surfaces where the\ngeometric structure is modelled on the complex plane up to affine maps whose\nlinear part is real. They are the geometric framework to study suspensions of\naffine interval exchange maps. However, though the SL(2,\ℝ)-action is\nergodic in connected components of strata of translation surfaces, there may be\nmutually disjoint SL(2,\ℝ)-invariant open subsets in components of\ndilation surfaces. A first distinction is between triangulable and\nnon-triangulable dilation surfaces. For non-triangulable surfaces, the action\nof SL(2,\ℝ) is somewhat trivial so the study can be focused on the\nspace of triangulable dilation surfaces. newline In this paper, we introduce\nthe notion of horizon saddle connections in order to refine the distinction\nbetween triangulable and non-triangulable dilation surfaces. We also introduce\nthe family of quasi-Hopf surfaces that can be triangulable but display the same\ntrivial behavior as non-triangulable surfaces. We prove that existence of\nhorizon saddle connections drastically restricts the Veech group a dilation\nsurface can have.\n