Minimality of the Ehrenfest wind-tree model

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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

Abstract
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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:

[1]	A. Avila and P. Hubert, Recurrence for the Wind-Tree Model,, Annales de l'Institut Henri Poincaré - Analyse non linéaire, (). Google Scholar
[2]	A. S. Besicovitch, A problem on topological transformations of the plane. II.,, Proc. Cambridge Philos. Soc., 47 (1951), 38. doi: 10.1017/S0305004100026347. Google Scholar
[3]	C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?,, Chaos, 19 (2009). doi: 10.1063/1.3085954. Google Scholar
[4]	M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons,, Trans. Am. Math. Soc., 350* (1998), 3523. doi: 10.1090/S0002-9947-98-02089-3. Google Scholar*
[5]	V. Delecroix, Divergent trajectories in the periodic wind-tree model,, J. Mod. Dyn., 7 (2013), 1. doi: 10.3934/jmd.2013.7.1. Google Scholar
[6]	V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, Ann. Sci. ENS, 47 (2014), 1085. Google Scholar
[7]	C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Statistical mechanics: Microscopic chaos from brownian motion?,, Nature, 401* (1999). doi: 10.1038/44759. Google Scholar*
[8]	P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, Encykl. d. Math. Wissensch. IV 2 II, (1912), 10. Google Scholar
[9]	K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces,, Invent. Math., 197* (2014), 241. doi: 10.1007/s00222-013-0482-z. Google Scholar*
[10]	G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models,, Phys. Rev., 185* (1969), 308. doi: 10.1103/PhysRev.185.308. Google Scholar*
[11]	W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. AMS, 50 (1944), 915. doi: 10.1090/S0002-9904-1944-08262-1. Google Scholar
[12]	J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, J. Math. Phys., 21 (1980), 1802. doi: 10.1063/1.524633. Google Scholar
[13]	E. H. Hauge and E. G. D. Cohen, Normal and abnormal diffusion in Ehrenfest's wind-tree model,, J. Math. Phys., 10 (1969), 397. Google Scholar
[14]	P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, Compos. Math., 149* (2013), 1364. doi: 10.1112/S0010437X12000887. Google Scholar*
[15]	P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. doi: 10.3934/dcds.2013.33.4341. Google Scholar
[16]	P. Hubert, Pascal, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion,, J. Reine Angew. Math., 656* (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar*
[17]	A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons,, Math. Notes, 18 (1975), 291. Google Scholar
[18]	M. Keane, Interval exchange transformations,, Math. Z., 141* (1975), 25. doi: 10.1007/BF01236981. Google Scholar*
[19]	A. Málaga Sabogal, Étude D'une Famille de Transformations Préservant la Mesure de $\mathbbZ \times \mathbbT$,, Thèse Paris 11, (2014). Google Scholar
[20]	S. Marmi, P. Moussa and Y.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps,, J. AMS, 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar
[21]	H. Masur and S. Tabachnikov, Rational billiards and flat structures,, Handbook of dynamical systems, 1 (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar
[22]	D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, J. Mod. Dyn., 6 (2012), 477. Google Scholar
[23]	S. Troubetzkoy, Approximation and billiards,, Dynamical systems and Diophantine approximation, (2009), 173. Google Scholar
[24]	S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, J. Stat. Phys., 141* (2010), 60. doi: 10.1007/s10955-010-0026-5. Google Scholar*
[25]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Inventiones Mathematicae, 97 (1989), 553. doi: 10.1007/BF01388890. Google Scholar
[26]	Y. Vorobets, Periodic geodesics on translation surfaces,, Algebraic and topological dynamics, (2005), 205. doi: 10.1090/conm/385/07199. Google Scholar
[27]	H. Van Beyeren and E. H. Hauge, Abnormal diffusion in Ehrenfest's wind-tree model,, Physics Letters A, 39 (1972), 397. doi: 10.1016/0375-9601(72)90112-0. Google Scholar
[28]	W. Wood and F. Lado, Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest's wind-tree model,, J. Comp. Physics, 7 (1971), 528. doi: 10.1016/0021-9991(71)90109-4. Google Scholar

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]	A. S. Besicovitch, A problem on topological transformations of the plane. II.,, Proc. Cambridge Philos. Soc., 47 (1951), 38. doi: 10.1017/S0305004100026347. Google Scholar
[3]	C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?,, Chaos, 19 (2009). doi: 10.1063/1.3085954. Google Scholar
[4]	M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons,, Trans. Am. Math. Soc., 350* (1998), 3523. doi: 10.1090/S0002-9947-98-02089-3. Google Scholar*
[5]	V. Delecroix, Divergent trajectories in the periodic wind-tree model,, J. Mod. Dyn., 7 (2013), 1. doi: 10.3934/jmd.2013.7.1. Google Scholar
[6]	V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model,, Ann. Sci. ENS, 47 (2014), 1085. Google Scholar
[7]	C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Statistical mechanics: Microscopic chaos from brownian motion?,, Nature, 401* (1999). doi: 10.1038/44759. Google Scholar*
[8]	P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik,, Encykl. d. Math. Wissensch. IV 2 II, (1912), 10. Google Scholar
[9]	K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces,, Invent. Math., 197* (2014), 241. doi: 10.1007/s00222-013-0482-z. Google Scholar*
[10]	G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models,, Phys. Rev., 185* (1969), 308. doi: 10.1103/PhysRev.185.308. Google Scholar*
[11]	W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties,, Bull. AMS, 50 (1944), 915. doi: 10.1090/S0002-9904-1944-08262-1. Google Scholar
[12]	J. Hardy and J. Weber, Diffusion in a periodic wind-tree model,, J. Math. Phys., 21 (1980), 1802. doi: 10.1063/1.524633. Google Scholar
[13]	E. H. Hauge and E. G. D. Cohen, Normal and abnormal diffusion in Ehrenfest's wind-tree model,, J. Math. Phys., 10 (1969), 397. Google Scholar
[14]	P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, Compos. Math., 149* (2013), 1364. doi: 10.1112/S0010437X12000887. Google Scholar*
[15]	P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase,, Discrete Contin. Dyn. Syst., 33 (2013), 4341. doi: 10.3934/dcds.2013.33.4341. Google Scholar
[16]	P. Hubert, Pascal, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion,, J. Reine Angew. Math., 656* (2011), 223. doi: 10.1515/CRELLE.2011.052. Google Scholar*
[17]	A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons,, Math. Notes, 18 (1975), 291. Google Scholar
[18]	M. Keane, Interval exchange transformations,, Math. Z., 141* (1975), 25. doi: 10.1007/BF01236981. Google Scholar*
[19]	A. Málaga Sabogal, Étude D'une Famille de Transformations Préservant la Mesure de $\mathbbZ \times \mathbbT$,, Thèse Paris 11, (2014). Google Scholar
[20]	S. Marmi, P. Moussa and Y.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps,, J. AMS, 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar
[21]	H. Masur and S. Tabachnikov, Rational billiards and flat structures,, Handbook of dynamical systems, 1 (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar
[22]	D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces,, J. Mod. Dyn., 6 (2012), 477. Google Scholar
[23]	S. Troubetzkoy, Approximation and billiards,, Dynamical systems and Diophantine approximation, (2009), 173. Google Scholar
[24]	S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model,, J. Stat. Phys., 141* (2010), 60. doi: 10.1007/s10955-010-0026-5. Google Scholar*
[25]	W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Inventiones Mathematicae, 97 (1989), 553. doi: 10.1007/BF01388890. Google Scholar
[26]	Y. Vorobets, Periodic geodesics on translation surfaces,, Algebraic and topological dynamics, (2005), 205. doi: 10.1090/conm/385/07199. Google Scholar
[27]	H. Van Beyeren and E. H. Hauge, Abnormal diffusion in Ehrenfest's wind-tree model,, Physics Letters A, 39 (1972), 397. doi: 10.1016/0375-9601(72)90112-0. Google Scholar
[28]	W. Wood and F. Lado, Monte Carlo calculation of normal and abnormal diffusion in Ehrenfest's wind-tree model,, J. Comp. Physics, 7 (1971), 528. doi: 10.1016/0021-9991(71)90109-4. Google Scholar

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