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

Divergent trajectories in the periodic wind-tree model

Vincent Delecroix 1,

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.
Keywords: Wind-tree model, rational billiard, interval-exchange transformations..
Mathematics Subject Classification: Primary: 37B05, 37B1.
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]

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V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic the wind-tree model,, preprint, ().   Google Scholar

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V. Delecroix and C. Ulcigrai, Diagonal changes in hyperelliptic strata. A natural extension to Ferenczi-Zamboni induction,, preprint., ().   Google Scholar

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S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392.  Google Scholar

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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.  Google Scholar

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J. Hardy and J. Weber, Diffusion in a periodic wind-tree model, J. Math. Phys., 21 (1980), 1802-1808. doi: 10.1063/1.524633.  Google Scholar

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A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.  Google Scholar

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M. Keane, Interval-exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

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G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

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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.  Google Scholar

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S. Tabachnikov, "Billards," Panoramas et Synthèses, Société Mathématiques de France, 1995. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[33]

M. Viana, Dynamics of interval-exchange maps and Teichmüller flows,, preprint. Available from: \url{http://w3.impa.br/~viana/out/ietf.pdf}., ().   Google Scholar

[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.  Google Scholar

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show all references

References:
[1]

A. Avila and P. Hubert, Recurrence for the windtree model,, preprint., ().   Google Scholar

[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.  Google Scholar

[3]

N. Chevallier and J.-P. Conze, Examples of recurrent or transient stationary walks in $\mathbbR^d$ over a rotation of $\mathbbT^2$, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 71-84. doi: 10.1090/conm/485/09493.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic the wind-tree model,, preprint, ().   Google Scholar

[7]

V. Delecroix and C. Ulcigrai, Diagonal changes in hyperelliptic strata. A natural extension to Ferenczi-Zamboni induction,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of interval-exchange transformations, Ann. Sci. Éc. Norm. Sup. (4), 44 (2011), 361-392.  Google Scholar

[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.  Google Scholar

[12]

K. Frączek and C. Ulcigrai, Non-ergodic $\ZZ$-periodic billiards and infinite translation surfaces,, preprint, ().   Google Scholar

[13]

, K. Frączek and C. Ulcigrai,, \textit{Ergodic directions for billiards in a strip with periodically located obstacles}, ().   Google Scholar

[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.  Google Scholar

[15]

D. Hensley, "Continued Fractions," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774682.  Google Scholar

[16]

P. Hooper, The invariant measures of some infinite interval-exchange maps,, preprint, ().   Google Scholar

[17]

P. Hooper, P. Hubert and B. Weiss, Dynamics on the infinite stair case surface,, to appear in Dis. Cont. Dyn. Sys., ().   Google Scholar

[18]

P. Hooper and B. Weiss, Generalized staircases: Recurrence and symmetry, Ann. Inst. Fourier, 62 (2012), 1581-1600. doi: 10.5802/aif.2730.  Google Scholar

[19]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$, Israel J. Math., 151 (2006), 281-321. doi: 10.1007/BF02777365.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

, P. Hubert and C. Ulcigrai,, \emph{Private communication}., ().   Google Scholar

[23]

P. Hubert and B. Weiss, Ergodicity for infinite periodic translation surfaces,, preprint., ().   Google Scholar

[24]

A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian) Mat. Zametki, 18 (1975), 291-300.  Google Scholar

[25]

M. Keane, Interval-exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[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.  Google Scholar

[30]

S. Tabachnikov, "Billards," Panoramas et Synthèses, Société Mathématiques de France, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

M. Viana, Dynamics of interval-exchange maps and Teichmüller flows,, preprint. Available from: \url{http://w3.impa.br/~viana/out/ietf.pdf}., ().   Google Scholar

[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.  Google Scholar

[35]

W. Stein, et al., Sage Mathematics Software (Version 4.5.2),, 2009. Available from: \url{http://www.sagemath.org}., ().   Google Scholar

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