Recurrence in generic staircases

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March 2012, 32(3): 1047-1053. doi: 10.3934/dcds.2012.32.1047

Recurrence in generic staircases

Serge Troubetzkoy ^1,

1.	Aix-Marseille University, Centre de physique théorique, Fédération de Recherches des Unités de Mathématique de Marseille, and Institut de mathématiques de Luminy, Luminy, Case 907, F-13288 Marseille Cedex 9, France

Received October 2010 Revised October 2010 Published October 2011

Abstract
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The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.

Keywords: translation surface., Recurrence.

Mathematics Subject Classification: Primary: 37D40, 37D5.

Citation: Serge Troubetzkoy. Recurrence in generic staircases. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047

References:

[1]	G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes, preprint, arXiv:0909.3069.
[2]	E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards, in "International Conference of Dynamical Systems" (Montevideo, 1995) (eds. F. Ledrappier, et al.), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, (1996), 21-45.
[3]	W. P. Hooper, Dynamics on an infinite surface with the lattice property, arXiv:0802.0189.
[4]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, preprint, arXiv:0905.3736.
[5]	P. Hubert and B. Weiss, Dynamics on an infinite staircase, preprint, 2008.
[6]	P. Hubert and G. Schmithüsen, Infinite translation surface with infinitely generated Veech groups, 2009.
[7]	P. Hubert, S. Lelievre and S. Troubetzkoy, On the Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, preprint, to appear in Journal für die reine und angewandte Mathematik (Crelle's Journal), arXiv:arXiv:0912.2891.
[8]	S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311.
[9]	, M. Lenci and S. Troubetzkoy, in preparation.
[10]	Ǐ. Schmeling and S. Troubetzkoǐ, Inhomogeneous Diophantine approximation and angular recurrence for billiards in polygons, Sb. Math. 194 (2003), 295-309. doi: 10.1070/SM2003v194n02ABEH000717.
[11]	K. Schmidt, "Cocyles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1., Macmillan Company on India, Ltd., Delhi, 1977.
[12]	S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn., 9 (2004), 1-12. doi: 10.1070/RD2004v009n01ABEH000259.
[13]	S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, Journal of Statistical Physics, 141 (2010), 60-67. doi: 10.1007/s10955-010-0026-5.
[14]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

show all references

References:

[1]	G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes, preprint, arXiv:0909.3069.
[2]	E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards, in "International Conference of Dynamical Systems" (Montevideo, 1995) (eds. F. Ledrappier, et al.), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, (1996), 21-45.
[3]	W. P. Hooper, Dynamics on an infinite surface with the lattice property, arXiv:0802.0189.
[4]	W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, preprint, arXiv:0905.3736.
[5]	P. Hubert and B. Weiss, Dynamics on an infinite staircase, preprint, 2008.
[6]	P. Hubert and G. Schmithüsen, Infinite translation surface with infinitely generated Veech groups, 2009.
[7]	P. Hubert, S. Lelievre and S. Troubetzkoy, On the Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, preprint, to appear in Journal für die reine und angewandte Mathematik (Crelle's Journal), arXiv:arXiv:0912.2891.
[8]	S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311.
[9]	, M. Lenci and S. Troubetzkoy, in preparation.
[10]	Ǐ. Schmeling and S. Troubetzkoǐ, Inhomogeneous Diophantine approximation and angular recurrence for billiards in polygons, Sb. Math. 194 (2003), 295-309. doi: 10.1070/SM2003v194n02ABEH000717.
[11]	K. Schmidt, "Cocyles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1., Macmillan Company on India, Ltd., Delhi, 1977.
[12]	S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn., 9 (2004), 1-12. doi: 10.1070/RD2004v009n01ABEH000259.
[13]	S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, Journal of Statistical Physics, 141 (2010), 60-67. doi: 10.1007/s10955-010-0026-5.
[14]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

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