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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

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 & 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, ().   Google Scholar

[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.  Google Scholar

[3]

W. P. Hooper, Dynamics on an infinite surface with the lattice property,, \arXiv{0802.0189}., ().   Google Scholar

[4]

W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, preprint, ().   Google Scholar

[5]

P. Hubert and B. Weiss, Dynamics on an infinite staircase, preprint, 2008. Google Scholar

[6]

P. Hubert and G. Schmithüsen, Infinite translation surface with infinitely generated Veech groups,, 2009., ().   Google Scholar

[7]

P. Hubert, S. Lelievre and S. Troubetzkoy, On the Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, preprint, ().   Google Scholar

[8]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311.  Google Scholar

[9]

, M. Lenci and S. Troubetzkoy,, in preparation., ().   Google Scholar

[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.  Google Scholar

[11]

K. Schmidt, "Cocyles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1., Macmillan Company on India, Ltd., Delhi, 1977. Google Scholar

[12]

S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn., 9 (2004), 1-12. doi: 10.1070/RD2004v009n01ABEH000259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes,, preprint, ().   Google Scholar

[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.  Google Scholar

[3]

W. P. Hooper, Dynamics on an infinite surface with the lattice property,, \arXiv{0802.0189}., ().   Google Scholar

[4]

W. P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry,, preprint, ().   Google Scholar

[5]

P. Hubert and B. Weiss, Dynamics on an infinite staircase, preprint, 2008. Google Scholar

[6]

P. Hubert and G. Schmithüsen, Infinite translation surface with infinitely generated Veech groups,, 2009., ().   Google Scholar

[7]

P. Hubert, S. Lelievre and S. Troubetzkoy, On the Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion,, preprint, ().   Google Scholar

[8]

S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math., 124 (1986), 293-311.  Google Scholar

[9]

, M. Lenci and S. Troubetzkoy,, in preparation., ().   Google Scholar

[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.  Google Scholar

[11]

K. Schmidt, "Cocyles on Ergodic Transformation Groups," Macmillan Lectures in Mathematics, Vol. 1., Macmillan Company on India, Ltd., Delhi, 1977. Google Scholar

[12]

S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn., 9 (2004), 1-12. doi: 10.1070/RD2004v009n01ABEH000259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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