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Conformal Geometry and Dynamics

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Veech surfaces and their periodic points


Author: Yoshihiko Shinomiya
Journal: Conform. Geom. Dyn. 20 (2016), 176-196
MSC (2010): Primary 30F35; Secondary 32G15, 32G05, 32G08
DOI: https://doi.org/10.1090/ecgd/296
Published electronically: May 19, 2016
MathSciNet review: 3513566
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Abstract: We give inequalities comparing widths or heights of cylinder decompositions of Veech surfaces with the signatures of their Veech groups. As an application of these inequalities, we estimate the numbers of periodic points of non-arithmetic Veech surfaces. The upper bounds depend only on the topological types of Veech surfaces and the signatures of Veech groups as Fuchsian groups. The upper bounds also estimate the numbers of holomorphic sections of holomorphic families of Riemann surfaces constructed from Veech groups of non-arithmetic Veech surfaces.


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Additional Information

Yoshihiko Shinomiya
Affiliation: Mathematics Education, Faculty of Education College of Education, Academic Institute 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
Email: shinomiya.yoshihiko@shizuoka.ac.jp

DOI: https://doi.org/10.1090/ecgd/296
Keywords: Flat surfaces, Veech surfaces, Veech groups, periodic points, holomorphic families of Riemann surfaces.
Received by editor(s): July 27, 2015
Received by editor(s) in revised form: November 23, 2015, and March 17, 2016
Published electronically: May 19, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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