Counting problem on wind-tree models

Pardo, Angel¹

¹ Institut de Mathématiques de Marseille, Aix-Marseille Université, Marseille, France

Geometry & topology, Tome 22 (2018) no. 3, pp. 1483-1536.

Résumé

We study periodic wind-tree models, that is, billiards in the plane endowed with ${\mathbb{Z}}^2$–periodically located identical connected symmetric right-angled obstacles. We give asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to ${\mathbb{Z}}^2$–translations) on the wind-tree billiard. We also explicitly compute the associated Siegel–Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

Détail

DOI : 10.2140/gt.2018.22.1483

Classification : 37D50, 37C35, 30F30, 37A40, 37D40
Keywords: billiards, translations surfaces, periodic orbits, counting problem, Siegel–Veech constants

Bibliographie

[1] J S Athreya, A Eskin, A Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on CP1, Ann. Sci. Éc. Norm. Supér. 49 (2016) 1311 | DOI

[2] A Avila, P Hubert, Recurrence for the wind-tree model, (2013)

[3] C Boissy, Configurations of saddle connections of quadratic differentials on CP1 and on hyperelliptic Riemann surfaces, Comment. Math. Helv. 84 (2009) 757 | DOI

[4] J Chaika, A Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn. 9 (2015) 1 | DOI

[5] N Chevallier, J P Conze, Examples of recurrent or transient stationary walks in Rd over a rotation of T2, from: "Ergodic theory" (editor I Assani), Contemp. Math. 485, Amer. Math. Soc. (2009) 71 | DOI

[6] V Delecroix, Divergent trajectories in the periodic wind-tree model, J. Mod. Dyn. 7 (2013) 1 | DOI

[7] V Delecroix, P Hubert, S Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. 47 (2014) 1085 | DOI

[8] V Delecroix, A Zorich, Cries and whispers in wind-tree forests, preprint (2015)

[9] P Ehrenfest, T Ehrenfest, Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, IV 2 II, Heft 6, Art. 32 (1912)

[10] A Eskin, M Kontsevich, A Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014) 207 | DOI

[11] A Eskin, J Marklof, D Witte Morris, Unipotent flows on the space of branched covers of Veech surfaces, Ergodic Theory Dynam. Systems 26 (2006) 129 | DOI

[12] A Eskin, H Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001) 443 | DOI

[13] A Eskin, H Masur, M Schmoll, Billiards in rectangles with barriers, Duke Math. J. 118 (2003) 427 | DOI

[14] A Eskin, H Masur, A Zorich, Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel–Veech constants, Publ. Math. Inst. Hautes Études Sci. 97 (2003) 61 | DOI

[15] A Eskin, M Mirzakhani, Invariant and stationary measures for the SL(2, R) action on moduli space, preprint (2013)

[16] A Eskin, M Mirzakhani, A Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2, R) action on moduli space, Ann. of Math. 182 (2015) 673 | DOI

[17] G Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle, J. Mod. Dyn. 5 (2011) 355 | DOI

[18] G Forni, C Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn. 8 (2014) 271 | DOI

[19] G Forni, C Matheus, A Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Systems 34 (2014) 353 | DOI

[20] K Frączek, C Ulcigrai, Non-ergodic Z–periodic billiards and infinite translation surfaces, Invent. Math. 197 (2014) 241 | DOI

[21] E Goujard, Siegel–Veech constants for strata of moduli spaces of quadratic differentials, Geom. Funct. Anal. 25 (2015) 1440 | DOI

[22] H W Gould, Combinatorial identities, Henry W Gould (1972)

[23] E Gutkin, C Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000) 191 | DOI

[24] J Hardy, J Weber, Diffusion in a periodic wind-tree model, J. Math. Phys. 21 (1980) 1802 | DOI

[25] H Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1982) 169 | DOI

[26] H Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, from: "Holomorphic functions and moduli, I" (editors D Drasin, C J Earle, F W Gehring, I Kra, A Marden), Math. Sci. Res. Inst. Publ. 10, Springer (1988) 215 | DOI

[27] H Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems 10 (1990) 151 | DOI

[28] H Masur, A Zorich, Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal. 18 (2008) 919 | DOI

[29] M Schmoll, On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori, Geom. Funct. Anal. 12 (2002) 622 | DOI

[30] W A Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. 115 (1982) 201 | DOI

[31] W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553 | DOI

[32] W A Veech, The billiard in a regular polygon, Geom. Funct. Anal. 2 (1992) 341 | DOI

[33] W A Veech, Siegel measures, Ann. of Math. 148 (1998) 895 | DOI

[34] Y B Vorobets, Ergodicity of billiards in polygons, Mat. Sb. 188 (1997) 65

[35] Y Vorobets, Periodic geodesics on generic translation surfaces, from: "Algebraic and topological dynamics" (editors S Kolyada, Y Manin, T Ward), Contemp. Math. 385, Amer. Math. Soc. (2005) 205 | DOI

[36] A Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015) 63 | DOI

[37] A Zorich, Flat surfaces, from: "Frontiers in number theory, physics, and geometry, I" (editors P Cartier, B Julia, P Moussa, P Vanhove), Springer (2006) 437 | DOI