Divergent trajectories in the periodic wind-tree model

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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

Abstract
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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]	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 $\mathbbR^d$ over a rotation of $\mathbbT^2$, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 71-84. doi: 10.1090/conm/485/09493.
[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, ().
[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, ().
[13]	, K. Frączek and C. Ulcigrai,, \textit{Ergodic directions for billiards in a strip with periodically located obstacles}, ().
[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, ().
[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,, \emph{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: \url{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: \url{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 $\mathbbR^d$ over a rotation of $\mathbbT^2$, in "Ergodic Theory," Contemp. Math., 485, Amer. Math. Soc., (2009), 71-84. doi: 10.1090/conm/485/09493.
[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, ().
[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, ().
[13]	, K. Frączek and C. Ulcigrai,, \textit{Ergodic directions for billiards in a strip with periodically located obstacles}, ().
[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, ().
[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,, \emph{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: \url{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: \url{http://www.sagemath.org}., ().

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