Veech groups, irrational billiards and stable abelian differentials

Ferrán Valdez

doi:10.3934/dcds.2012.32.1055

Article Contents

2012, Volume 32, Issue 3: 1055-1063. Doi: 10.3934/dcds.2012.32.1055

This issue Previous Article Recurrence in generic staircases Next Article The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay

Veech groups, irrational billiards and stable abelian differentials

Ferrán Valdez^1,

1.
Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58089, Morelia, Michoacán

Received: April 30, 2010

Revised: June 30, 2011

Published: February 2012

Abstract Related Papers Cited by

Abstract

We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of $\rm SO(2,\mathbf{R})$ and we calculate their rank.

Keywords:

Polygonal Billiards,
Veech group.

Mathematics Subject Classification: Primary: 37D50; Secondary: 37J35.

Citation:

Related Papers

Cited by

References

[1]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073.doi: 10.2140/gt.2007.11.1887.
[2]	É. Ghys, Topologie des feuilles génériques, Ann. of Math. (2), 141 (1995), 387-422.doi: 10.2307/2118526.
[3]	E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards, in "International Conference on Dynamical Systems" (Montevideo, 1995), 21-45, Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996.
[4]	P. Hooper, Dynamics on an infinite surface with the lattice property, preprint, arXiv:0802.0189.
[5]	P. Hooper and B. Weiss, Generalized staircases: recurrence and symmetry, To appear in Annales de l'Institut Fourier, 2009.
[6]	P Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase surface, To appear in Dis. Cont. Dyn. Sys., 2010.
[7]	P. Hubert and G. Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, Journal of Modern Dynamics, 4 (2010), 715-732.doi: 10.3934/jmd.2010.4.715.
[8]	A. B. Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard, Comm. Math. Phys., 111 (1987), 151-160.doi: 10.1007/BF01239021.
[9]	H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, 1015-1089, North Holland, Amsterdam, 2002.
[10]	P. Przytycki, F. Valdez and G. Weitze-Schmithüsen, Veech groups of Loch Ness monsters, To appear in Annales de l'Institut Fourier, 2009.
[11]	F. Valdez, Infinite genus surfaces and irrational polygonal billiards, Geom. Dedicata, 143 (2009), 143-154.doi: 10.1007/s10711-009-9378-x.
[12]	S. Tabachnikov, Billiards, Panor. Synth. No., 1 (1995), vi+142 pp.
[13]	W. A. Veech, Teichmüller curves in the moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553-583.doi: 10.1007/BF01388890.