American Institute of Mathematical Sciences

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2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209

Minimality of the Ehrenfest wind-tree model

Alba Málaga Sabogal 1, and  Serge Troubetzkoy 1,

1. 

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France, France

Received  June 2015 Revised  March 2016 Published  June 2016

We consider aperiodic wind-tree models and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions and has a dense set of periodic points.
Keywords: minimality, periodic orbit., Wind-tree.
Mathematics Subject Classification: Primary: 37D50; Secondary: 37A40, 37B0.
Citation: Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209
References:
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show all references

References:
[1]

A. Avila and P. Hubert, Recurrence for the Wind-Tree Model,, Annales de l'Institut Henri Poincaré - Analyse non linéaire, ().   Google Scholar

[2]

Proc. Cambridge Philos. Soc., 47 (1951), 38-45. doi: 10.1017/S0305004100026347.  Google Scholar

[3]

Chaos, 19 (2009), 013121, 10pp. doi: 10.1063/1.3085954.  Google Scholar

[4]

Trans. Am. Math. Soc., 350 (1998), 3523-3535. doi: 10.1090/S0002-9947-98-02089-3.  Google Scholar

[5]

J. Mod. Dyn., 7 (2013), 1-29. doi: 10.3934/jmd.2013.7.1.  Google Scholar

[6]

Ann. Sci. ENS, 47 (2014), 1085-1110.  Google Scholar

[7]

Nature, 401 (1999), p875. doi: 10.1038/44759.  Google Scholar

[8]

Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (in German, translated in:) The conceptual foundations of the statistical approach in mechanics, (trans. Moravicsik, M. J.), 10-13 Cornell University Press, Itacha NY (1959). Google Scholar

[9]

Invent. Math., 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z.  Google Scholar

[10]

Phys. Rev., 185 (1969), 308-322. doi: 10.1103/PhysRev.185.308.  Google Scholar

[11]

Bull. AMS, 50 (1944), 915-919. doi: 10.1090/S0002-9904-1944-08262-1.  Google Scholar

[12]

J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.  Google Scholar

[13]

J. Math. Phys., 10 (1969), 397-414. Google Scholar

[14]

Compos. Math., 149 (2013), 1364-1380. doi: 10.1112/S0010437X12000887.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 33 (2013), 4341-4347. doi: 10.3934/dcds.2013.33.4341.  Google Scholar

[16]

J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052.  Google Scholar

[17]

Math. Notes, 18 (1975), 291-300.  Google Scholar

[18]

Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[19]

Thèse Paris 11, 2014. Google Scholar

[20]

J. AMS, 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X.  Google Scholar

[21]

Handbook of dynamical systems, North-Holland, Amsterdam, 1 (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar

[22]

J. Mod. Dyn., 6 (2012), 477-497.  Google Scholar

[23]

Dynamical systems and Diophantine approximation, 173-185, Semin. Congr., 19, Soc. Math. France, Paris, 2009.  Google Scholar

[24]

J. Stat. Phys., 141 (2010), 60-67. doi: 10.1007/s10955-010-0026-5.  Google Scholar

[25]

Inventiones Mathematicae, 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

[26]

Algebraic and topological dynamics, 205-258, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005 doi: 10.1090/conm/385/07199.  Google Scholar

[27]

Physics Letters A, 39 (1972), 397-398. doi: 10.1016/0375-9601(72)90112-0.  Google Scholar

[28]

J. Comp. Physics, 7 (1971), 528-546. doi: 10.1016/0021-9991(71)90109-4.  Google Scholar

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