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. 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, 485 (2009), 71. doi: 10.1090/conm/485/09493.

[4]

J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. 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. 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, (1959).

[9]

S. Ferenczi and L. Zamboni, Structure of $K$-interval-exchange transformations: Induction trajectories, and distance theorems,, J. Anal. Math., 112 (2010), 289. 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.

[11]

R. Fox and R. Kershner, Concerning the transitive properties of geodesics in rational polyhedron,, Duke Math. J., 2 (1936), 147. 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. doi: 10.1063/1.524633.

[15]

D. Hensley, "Continued Fractions,", World Scientific Publishing Co. Pte. Ltd., (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. doi: 10.5802/aif.2730.

[19]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Israel J. Math., 151 (2006), 281. 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. 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. 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.

[25]

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

[26]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. 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. doi: 10.1090/S0894-0347-03-00432-6.

[28]

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

[29]

J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon,, Proc. Lond. Math. Soc., 102 (2011), 291. doi: 10.1112/plms/pdq018.

[30]

S. Tabachnikov, "Billards,", Panoramas et Synthèses, (1995).

[31]

W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. 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. 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, (2006), 437. 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. 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, 485 (2009), 71. doi: 10.1090/conm/485/09493.

[4]

J.-P. Conze, Recurrence, ergodicity and invariant measures for cocycles over a rotation,, in, 485 (2009), 45. 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. 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, (1959).

[9]

S. Ferenczi and L. Zamboni, Structure of $K$-interval-exchange transformations: Induction trajectories, and distance theorems,, J. Anal. Math., 112 (2010), 289. 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.

[11]

R. Fox and R. Kershner, Concerning the transitive properties of geodesics in rational polyhedron,, Duke Math. J., 2 (1936), 147. 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. doi: 10.1063/1.524633.

[15]

D. Hensley, "Continued Fractions,", World Scientific Publishing Co. Pte. Ltd., (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. doi: 10.5802/aif.2730.

[19]

P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in $\mathcalH(2)$,, Israel J. Math., 151 (2006), 281. 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. 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. 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.

[25]

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

[26]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. 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. doi: 10.1090/S0894-0347-03-00432-6.

[28]

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

[29]

J. Smillie and C. Ulcigrai, Beyond Sturmian sequences: Coding linear trajectories in the regular octagon,, Proc. Lond. Math. Soc., 102 (2011), 291. doi: 10.1112/plms/pdq018.

[30]

S. Tabachnikov, "Billards,", Panoramas et Synthèses, (1995).

[31]

W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. 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. 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, (2006), 437. 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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