American Institute of Mathematical Sciences

Advanced
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 - A, 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, 362 (1996), 21.   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.   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. \textbf{194} (2003), 194 (2003), 295.  doi: 10.1070/SM2003v194n02ABEH000717.  Google Scholar

[11]

K. Schmidt, "Cocyles on Ergodic Transformation Groups,", Macmillan Lectures in Mathematics, (1977).   Google Scholar

[12]

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

[13]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, Journal of Statistical Physics, 141 (2010), 60.  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.  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, 362 (1996), 21.   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.   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. \textbf{194} (2003), 194 (2003), 295.  doi: 10.1070/SM2003v194n02ABEH000717.  Google Scholar

[11]

K. Schmidt, "Cocyles on Ergodic Transformation Groups,", Macmillan Lectures in Mathematics, (1977).   Google Scholar

[12]

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

[13]

S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, Journal of Statistical Physics, 141 (2010), 60.  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.  doi: 10.1007/BF01388890.  Google Scholar

[1]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[2]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[3]

José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[4]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[5]

Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[6]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[7]

Milton Ko. Rényi entropy and recurrence. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403
[8]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[9]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[10]

Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059
[11]

Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
[12]

A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587
[13]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715
[14]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[15]

Vincent Penné, Benoît Saussol, Sandro Vaienti. Dimensions for recurrence times: topological and dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 783-798. doi: 10.3934/dcds.1999.5.783
[16]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[17]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[18]

L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821
[19]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[20]

Lok Ming Lui, Chengfeng Wen, Xianfeng Gu. A conformal approach for surface inpainting. Inverse Problems & Imaging, 2013, 7 (3) : 863-884. doi: 10.3934/ipi.2013.7.863

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences