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Journal of Modern Dynamics (JMD)
 

Minimality of the Ehrenfest wind-tree model
Pages: 209 - 228, Volume 10, 2016

doi:10.3934/jmd.2016.10.209      Abstract        References        Full text (233.6K)           Related Articles

Alba Málaga Sabogal - Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France (email)
Serge Troubetzkoy - Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France (email)

1 A. Avila and P. Hubert, Recurrence for the Wind-Tree Model, Annales de l'Institut Henri Poincaré - Analyse non linéaire, to appear.
2 A. S. Besicovitch, A problem on topological transformations of the plane. II., Proc. Cambridge Philos. Soc., 47 (1951), 38-45.       
3 C. Bianca and L. Rondoni, The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles?, Chaos, 19 (2009), 013121, 10pp.       
4 M. Boshernitzan, G. Galperin, T. Krüger and S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons, Trans. Am. Math. Soc., 350 (1998), 3523-3535.       
5 V. Delecroix, Divergent trajectories in the periodic wind-tree model, J. Mod. Dyn., 7 (2013), 1-29.       
6 V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. ENS, 47 (2014), 1085-1110.       
7 C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Statistical mechanics: Microscopic chaos from brownian motion?, Nature, 401 (1999), p875.
8 P. and T. Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, 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).
9 K. Frączek and C. Ulcigrai, Non-ergodic $\mathbbZ$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298.       
10 G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models, Phys. Rev., 185 (1969), 308-322.
11 W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. AMS, 50 (1944), 915-919.       
12 J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808.       
13 E. H. Hauge and E. G. D. Cohen, Normal and abnormal diffusion in Ehrenfest's wind-tree model, J. Math. Phys., 10 (1969), 397-414.
14 P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, Compos. Math., 149 (2013), 1364-1380.       
15 P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite staircase, Discrete Contin. Dyn. Syst., 33 (2013), 4341-4347.       
16 P. Hubert, Pascal, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, J. Reine Angew. Math., 656 (2011), 223-244.       
17 A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Math. Notes, 18 (1975), 291-300.       
18 M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.       
19 A. Málaga Sabogal, Étude D'une Famille de Transformations Préservant la Mesure de $\mathbbZ \times \mathbbT$, Thèse Paris 11, 2014.
20 S. Marmi, P. Moussa and Y.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. AMS, 18 (2005), 823-872.       
21 H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, North-Holland, Amsterdam, 1 (2002), 1015-1089.       
22 D. Ralston and S. Troubetzkoy, Ergodic infinite group extensions of geodesic flows on translation surfaces, J. Mod. Dyn., 6 (2012), 477-497.       
23 S. Troubetzkoy, Approximation and billiards, Dynamical systems and Diophantine approximation, 173-185, Semin. Congr., 19, Soc. Math. France, Paris, 2009.       
24 S. Troubetzkoy, Typical recurrence for the Ehrenfest wind-tree model, J. Stat. Phys., 141 (2010), 60-67.       
25 W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae, 97 (1989), 553-583.       
26 Y. Vorobets, Periodic geodesics on translation surfaces, Algebraic and topological dynamics, 205-258, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005       
27 H. Van Beyeren and E. H. Hauge, Abnormal diffusion in Ehrenfest's wind-tree model, Physics Letters A, 39 (1972), 397-398.
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), 528-546.       

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