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The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers placed at each integer point. We prove that independently of the size of scatters and generically with respect to the angle, the polynomial diffusion rate in this billiard is .
Le vent dans les arbres périodique est un billard infini construit de la manière suivante. On considère le plan dans lequel sont placés des obstacles rectangulaires identiques à chaque point entier. Une particule (identifiée à un point) se déplace en ligne droite (le vent) et rebondit de manière élastique sur les obstacles (les arbres). Nous prouvons qu'indépendamment de la taille des obstacles et génériquement par rapport à l'angle initial de la particule le coefficient de diffusion polynomial des orbites de ce billard est .
@article{ASENS_2014__47_6_1085_0,
author = {Delecroix, Vincent and Hubert, Pascal and Leli\`evre, Samuel},
title = {Diffusion for the periodic wind-tree model},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1085--1110},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 47},
number = {6},
year = {2014},
doi = {10.24033/asens.2234},
mrnumber = {3297155},
zbl = {1351.37159},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2234/}
}
TY - JOUR AU - Delecroix, Vincent AU - Hubert, Pascal AU - Lelièvre, Samuel TI - Diffusion for the periodic wind-tree model JO - Annales scientifiques de l'École Normale Supérieure PY - 2014 SP - 1085 EP - 1110 VL - 47 IS - 6 PB - Société Mathématique de France. Tous droits réservés UR - http://geodesic.mathdoc.fr/articles/10.24033/asens.2234/ DO - 10.24033/asens.2234 LA - en ID - ASENS_2014__47_6_1085_0 ER -
%0 Journal Article %A Delecroix, Vincent %A Hubert, Pascal %A Lelièvre, Samuel %T Diffusion for the periodic wind-tree model %J Annales scientifiques de l'École Normale Supérieure %D 2014 %P 1085-1110 %V 47 %N 6 %I Société Mathématique de France. Tous droits réservés %U http://geodesic.mathdoc.fr/articles/10.24033/asens.2234/ %R 10.24033/asens.2234 %G en %F ASENS_2014__47_6_1085_0
Delecroix, Vincent; Hubert, Pascal; Lelièvre, Samuel. Diffusion for the periodic wind-tree model. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 6, pp. 1085-1110. doi : 10.24033/asens.2234. http://geodesic.mathdoc.fr/articles/10.24033/asens.2234/
Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Volume 11 (2007), pp. 1887-2073 (ISSN: 1465-3060) | MR | Zbl | DOI
Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math., Volume 172 (2010), pp. 139-185 (ISSN: 0003-486X) | MR | Zbl | DOI
Limit theorems for translation flows, Ann. of Math., Volume 179 (2014), pp. 431-499 (ISSN: 0003-486X) | MR | Zbl | DOI
Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., Volume 17 (2004), pp. 871-908 (ISSN: 0894-0347) | MR | Zbl | DOI
Every flat surface is Birkhoff generic and Osceledets generic in almost every direction (preprint arXiv:13051104 ) | MR
On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Ergodic Theory Dynam. Systems, Volume 32 (2012), pp. 491-515 (ISSN: 0143-3857) | MR | Zbl | DOI
Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, Encykl. d. Math. Wissensch. IV 2 II, Heft 6 (1912) (in German, translated in:) The conceptual foundations of the statistical approach in mechanics, 10-13 Cornell University Press, Ithaca NY, (1959) | MR
Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011), pp. 319-353 (ISSN: 1930-5311) | MR | Zbl | DOI
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. I.H.É.S., Volume 120 (2014), pp. 207-333 (ISSN: 0073-8301) | MR | Zbl | mathdoc-id | DOI
Invariant and stationary measures for the action on moduli space (preprint arXiv:1302.3320 ) | MR
Isolation, Equidistribution, and Orbit Closures for the action on moduli space (preprint arXiv:1305.3015 ) | MR
Square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011), pp. 285-318 (ISSN: 1930-5311) | MR | Zbl | DOI
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., Volume 155 (2002), pp. 1-103 (ISSN: 0003-486X) | MR | Zbl | DOI
Dynamics on the infinite stair case surface (preprint arXiv:0905.3736, to appear in Dis. Cont. Dyn ) | MR | Zbl
The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, J. reine angew. Math., Volume 656 (2011), pp. 223-244 (ISSN: 0075-4102) | MR | Zbl | DOI
Infinite translation surfaces with infinitely generated Veech groups, J. Mod. Dyn., Volume 4 (2010), pp. 715-732 (ISSN: 1930-5311) | MR | Zbl | DOI
Generalized staircases: recurrence and symmetry, Ann. Inst. Fourier (Grenoble), Volume 62 (2012), pp. 1581-1600 http://aif.cedram.org/... (ISSN: 0373-0956) | MR | Zbl | mathdoc-id | DOI
Diffusion in a periodic wind-tree model, J. Math. Phys., Volume 21 (1980), pp. 1802-1808 (ISSN: 0022-2488) | DOI | MR
Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31 (ISSN: 0025-5874) | MR | Zbl | DOI
Lyapunov exponents and Hodge theory (preprint arXiv:hep-th/9701164 ) | MR | Zbl
Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003), pp. 631-678 (ISSN: 0020-9910) | MR | Zbl | DOI
Multi-geodesic tessellations, fractional Dehn twists and uniformization (preprint arXiv:math/0702374 )
Interval exchange transformations and measured foliations, Ann. of Math., Volume 115 (1982), pp. 169-200 (ISSN: 0003-486X) | MR | Zbl | DOI
Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 (ISSN: 0894-0347) | MR | Zbl | DOI
Teichmüller curves in genus two: discriminant and spin, Math. Ann., Volume 333 (2005), pp. 87-130 (ISSN: 0025-5831) | MR | Zbl | DOI
Dynamics of over moduli space in genus two, Ann. of Math., Volume 165 (2007), pp. 397-456 (ISSN: 0003-486X) | MR | Zbl | DOI
Veech groups of Loch Ness monsters, Ann. Inst. Fourier (Grenoble), Volume 61 (2011), pp. 673-687 (ISSN: 0373-0956) | MR | Zbl | mathdoc-id | DOI
Examples of origamis, The Geometry of Riemann Surfaces and Abelian Varieties (Contemp. Math.), Volume 397 (2006), pp. 193-206 | MR | Zbl | DOI
, Panor. Synth., 1, Soc. Math., France, 1995 | MR | Zbl
Veech groups, irrational billiards and stable abelian differentials, Discrete Contin. Dyn. Syst., Volume 32 (2012), pp. 1055-1063 (ISSN: 1078-0947) | MR | Zbl | DOI
Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982), pp. 201-242 (ISSN: 0003-486X) | MR | Zbl | DOI
Dynamics of interval exchange maps and Teichmüller flows (preprint http://w3.impa.br/~viana/out/ietf.pdf )
, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 401-435 | MR | Zbl | DOI
, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | MR | Zbl | DOI
Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), Volume 46 (1996), pp. 325-370 (ISSN: 0373-0956) | MR | Zbl | mathdoc-id | DOI
Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, Volume 17 (1997), pp. 1477-1499 (ISSN: 0143-3857) | MR | Zbl | DOI
, Pseudoperiodic topology (Amer. Math. Soc. Transl. Ser. 2), Volume 197, Amer. Math. Soc., Providence, RI, 1999, pp. 135-178 | MR | Zbl
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