# American Institute of Mathematical Sciences

January  2013, 7(1): 1-29. doi: 10.3934/jmd.2013.7.1

## Divergent trajectories in the periodic wind-tree model

 1 Université Paris 7, Département de Mathématiques, Bâtiment Sophie Germain, 8 Place FM/13, 75013 Paris, France

Received  April 2012 Published  May 2013

The periodic wind-tree model is a family $T(a,b)$ of billiards in the plane in which identical rectangular scatterers of size $a \times b$ are disposed periodically at each integer point. In that model, the recurrence is generic with respect to the parameters $a$, $b$, and the angle $\theta$ of initial direction of the particule. In contrast, we prove that for some parameters $(a,b)$ the set of angles $\theta$ for which the billiard flow is divergent has Hausdorff dimension greater than one half.
Citation: Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1
##### References:
 [1] A. Avila and P. Hubert, Recurrence for the windtree model, preprint. [2] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8. [3] N. Chevallier and J.-P. Conze, Examples of recurrent or transient stationary walks in $\mathbbmathbb{R}^{d}$ over a rotation of $\mathbbT^2$, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 71-84. doi: 10.1090/conm/485/09493. [4] J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 45-70. doi: 10.1090/conm/485/09492. [5] J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Erg. Th. and Dyn. Syst., 32, (2012), 491-515. doi: 10.1017/S0143385711001003. [6] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic the wind-tree model, preprint, arXiv:1107.1810v1. [7] V. Delecroix and C. Ulcigrai, Diagonal changes in hyperelliptic strata. A natural extension to Ferenczi-Zamboni induction, preprint. [8] P. Ehrenfeset and T. Ehrenfest, The conceptual foundations of the statistical approach in mechanics, Translated from the German by Michael J. Moravcsik, Reprint of the 1959 English edition, Dover Publications, Inc., New York, 1990. [9] S. Ferenczi and L. Zamboni, Structure of $K$-interval-exchange transformations: Induction trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2. [10] S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392. [11] R. Fox and R. Kershner, Concerning the transitive properties of geodesics in rational polyhedron, Duke Math. J., 2 (1936), 147-150. doi: 10.1215/S0012-7094-36-00213-2. [12] K. Frączek and C. Ulcigrai, Non-ergodic $\ZZ$-periodic billiards and infinite translation surfaces, preprint, arXiv:1109.4584v1. [13] , K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles, preprint arXiv:1208.5212. [14] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633. [15] D. Hensley, "Continued Fractions," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682. [16] P. Hooper, The invariant measures of some infinite interval-exchange maps, preprint, arXiv:1005.1902v1. [17] P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite stair case surface, to appear in Dis. Cont. Dyn. Sys. [18] P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, Ann. Inst. Fourier, 62 (2012), 1581-1600. doi: 10.5802/aif.2730. [19] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$, Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365. [20] P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052. [21] P. Hubert and G. Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn., 4 (2010), 715-732. doi: 10.3934/jmd.2010.4.715. [22] , P. Hubert and C. Ulcigrai, Private communication. [23] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, preprint. [24] A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300. [25] M. Keane, Interval-exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [26] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7. [27] C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 875-885. doi: 10.1090/S0894-0347-03-00432-6. [28] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [29] J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc., 102 (2011), 291-340. doi: 10.1112/plms/pdq018. [30] S. Tabachnikov, "Billards," Panoramas et Synthèses, Société Mathématiques de France, 1995. [31] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. [32] W. Veech, Teichmüller curves in moduli spaces, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. [33] M. Viana, Dynamics of interval-exchange maps and Teichmüller flows, preprint. Available from: http://w3.impa.br/~viana/out/ietf.pdf. [34] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. 1" Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13. [35] W. Stein, et al., Sage Mathematics Software (Version 4.5.2), 2009. Available from: http://www.sagemath.org.

show all references

##### References:
 [1] A. Avila and P. Hubert, Recurrence for the windtree model, preprint. [2] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8. [3] N. Chevallier and J.-P. Conze, Examples of recurrent or transient stationary walks in $\mathbbmathbb{R}^{d}$ over a rotation of $\mathbbT^2$, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 71-84. doi: 10.1090/conm/485/09493. [4] J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 45-70. doi: 10.1090/conm/485/09492. [5] J.-P. Conze and E. Gutkin, On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Erg. Th. and Dyn. Syst., 32, (2012), 491-515. doi: 10.1017/S0143385711001003. [6] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic the wind-tree model, preprint, arXiv:1107.1810v1. [7] V. Delecroix and C. Ulcigrai, Diagonal changes in hyperelliptic strata. A natural extension to Ferenczi-Zamboni induction, preprint. [8] P. Ehrenfeset and T. Ehrenfest, The conceptual foundations of the statistical approach in mechanics, Translated from the German by Michael J. Moravcsik, Reprint of the 1959 English edition, Dover Publications, Inc., New York, 1990. [9] S. Ferenczi and L. Zamboni, Structure of $K$-interval-exchange transformations: Induction trajectories, and distance theorems, J. Anal. Math., 112 (2010), 289-328. doi: 10.1007/s11854-010-0031-2. [10] S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392. [11] R. Fox and R. Kershner, Concerning the transitive properties of geodesics in rational polyhedron, Duke Math. J., 2 (1936), 147-150. doi: 10.1215/S0012-7094-36-00213-2. [12] K. Frączek and C. Ulcigrai, Non-ergodic $\ZZ$-periodic billiards and infinite translation surfaces, preprint, arXiv:1109.4584v1. [13] , K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles, preprint arXiv:1208.5212. [14] J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633. [15] D. Hensley, "Continued Fractions," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682. [16] P. Hooper, The invariant measures of some infinite interval-exchange maps, preprint, arXiv:1005.1902v1. [17] P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite stair case surface, to appear in Dis. Cont. Dyn. Sys. [18] P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, Ann. Inst. Fourier, 62 (2012), 1581-1600. doi: 10.5802/aif.2730. [19] P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$, Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365. [20] P. Hubert, S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, J. Reine Angew. Math., 656 (2011), 223-244. doi: 10.1515/CRELLE.2011.052. [21] P. Hubert and G. Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn., 4 (2010), 715-732. doi: 10.3934/jmd.2010.4.715. [22] , P. Hubert and C. Ulcigrai, Private communication. [23] P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces, preprint. [24] A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300. [25] M. Keane, Interval-exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [26] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089. doi: 10.1016/S1874-575X(02)80015-7. [27] C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 875-885. doi: 10.1090/S0894-0347-03-00432-6. [28] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [29] J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc., 102 (2011), 291-340. doi: 10.1112/plms/pdq018. [30] S. Tabachnikov, "Billards," Panoramas et Synthèses, Société Mathématiques de France, 1995. [31] W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. [32] W. Veech, Teichmüller curves in moduli spaces, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. [33] M. Viana, Dynamics of interval-exchange maps and Teichmüller flows, preprint. Available from: http://w3.impa.br/~viana/out/ietf.pdf. [34] A. Zorich, Flat surfaces, in "Frontiers in Number Theory, Physics, and Geometry. 1" Springer, Berlin, (2006), 437-583. doi: 10.1007/978-3-540-31347-2_13. [35] W. Stein, et al., Sage Mathematics Software (Version 4.5.2), 2009. Available from: http://www.sagemath.org.
 [1] 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 [2] Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123 [3] Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253 [4] Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139 [5] Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379 [6] Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010 [7] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [8] Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006 [9] Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53 [10] Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35 [11] Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003 [12] Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271 [13] Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251 [14] David Cowan. A billiard model for a gas of particles with rotation. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101 [15] Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042 [16] Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289 [17] Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077 [18] Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83 [19] Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180 [20] Sergei Avdonin, Jonathan Bell. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems and Imaging, 2015, 9 (3) : 645-659. doi: 10.3934/ipi.2015.9.645

2021 Impact Factor: 0.641