# American Institute of Mathematical Sciences

March  2012, 32(3): 1055-1063. doi: 10.3934/dcds.2012.32.1055

## Veech groups, irrational billiards and stable abelian differentials

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

Received  May 2010 Revised  July 2011 Published  October 2011

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.
Citation: Ferrán Valdez. Veech groups, irrational billiards and stable abelian differentials. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1055-1063. doi: 10.3934/dcds.2012.32.1055
##### 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, (). [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.

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##### 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, (). [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.
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