this context (Theorem 2.4). As an application we show a version of Margulis' theorem for the natural analogue of volume growth for translation surfaces (Theorem 6.11).This note originated as a summer MPhil project of the first author. It may have been possible to apply the transfer operator methods in [15], but instead we employ a more direct and elementary approach.We are grateful to A. Eskin, J. Chaika, R. Sharp, S. Ghazouani and the three anonymous referees for their useful comments. Infinite GraphsIn this section we will introduce the types of graphs we shall we working with as well as basic definitions which will be used throughout the paper.Let G be a non-empty connected oriented graph. Let V = V(G) and E = E(G) be the vertex and oriented edge sets respectively. For every edge e, let i(e) and t(e) denote the initial and the terminal vertex of e, respectively. We can define a length distance d on G by introducing a length function ℓ : E → R which assigns a positive real number ℓ(e) to each edge e ∈ E.Example 2.1 (Infinite Graph). Consider a graph G formed from one vertex and a countably infinite number of edges.
In this note we consider the entropy by Dankwart [On the large-scale geometry of flat surfaces, 2014, PhD thesis. https://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf] of unit area translation surfaces in the S L ( 2 , R ) SL(2, \mathbb R) orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices. We also provide a method for approximating the entropy of surfaces in the orbits.
In this note we consider the entropy [5] of unit area translation surfaces in the SL(2, R) orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices.The authors would like to thank L.Bétermin for suggesting the connection with his work [2]. The second author was supported by ERC Grant 833802-Resonances and EPSRC grant EP/T001674/1.1 This avoids the complication of accounting for cylinders of uncountably many parallel geodesics. Alternatively, we could account for these by counting only their free homotopy classes, but then their polynomial growth does not affect the definition of the entropy.
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