Right-angled billiards and volumes of moduli spaces of quadratic differentials on ℂP1
[Billards à angles droits et volumes des espaces de modules des différentielles quadratiques sur ℂP1 ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1311-1386.

Nous utilisons le lien entre les volumes des strates de différentielles méromorphes quadratiques avec des pôles simples sur ℂP1 et les fonctions de comptage du nombre de (cylindres de) géodésiques fermées simples pour la métrique plate associée afin de démontrer une formule très explicite pour le volume des strates, conjecturée par M. Kontsevich il y a une décennie.

En appliquant des techniques ergodiques au flot géodésique de Teichmüller nous obtenons une asymptotique quadratique pour le nombre de (bandes de) trajectoires fermées et le nombre de diagonales généralisées dans presque tout billard à angles « droits ».

We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on ℂP1 and counting functions of the number of (bands of) simple closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago.

Applying ergodic techniques to the Teichmüller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.

Publié le :
DOI : 10.24033/asens.2310
Classification : 32G15, 37E35, 37C27.
Keywords: Billiards, meromorphic differentials, moduli spaces, Siegel-Veech constants.
Mot clés : Billards, différentielles méromorphes, espaces de modules, constantes de Siegel-Veech. 
@article{ASENS_2016__49_6_1311_0,
     author = {Athreya, Jayadev S. and Eskin, Alex and Zorich, Anton},
     title = {Right-angled billiards  and volumes of moduli spaces  of quadratic differentials on~${\mathbb {C}}\!\operatorname{P}^1$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1311--1386},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {6},
     year = {2016},
     doi = {10.24033/asens.2310},
     mrnumber = {3592359},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2310/}
}
TY  - JOUR
AU  - Athreya, Jayadev S.
AU  - Eskin, Alex
AU  - Zorich, Anton
TI  - Right-angled billiards  and volumes of moduli spaces  of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 1311
EP  - 1386
VL  - 49
IS  - 6
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2310/
DO  - 10.24033/asens.2310
LA  - en
ID  - ASENS_2016__49_6_1311_0
ER  - 
%0 Journal Article
%A Athreya, Jayadev S.
%A Eskin, Alex
%A Zorich, Anton
%T Right-angled billiards  and volumes of moduli spaces  of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 1311-1386
%V 49
%N 6
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2310/
%R 10.24033/asens.2310
%G en
%F ASENS_2016__49_6_1311_0
Athreya, Jayadev S.; Eskin, Alex; Zorich, Anton. Right-angled billiards  and volumes of moduli spaces  of quadratic differentials on ${\mathbb {C}}\!\operatorname{P}^1$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1311-1386. doi : 10.24033/asens.2310. http://www.numdam.org/articles/10.24033/asens.2310/

Athreya, J. S.; Eskin, A.; Zorich, A. Counting generalized Jenkins-Strebel differentials, Geom. Dedicata, Volume 170 (2014), pp. 195-217 (ISSN: 0046-5755) | DOI | MR | Zbl

Avila, A.; Gouëzel, S. Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math., Volume 178 (2013), pp. 385-442 (ISSN: 0003-486X) | DOI | MR | Zbl

Avila, A.; Gouëzel, S.; Yoccoz, J.-C. Exponential mixing for the Teichmüller flow, Publ. Math. IHÉS, Volume 104 (2006), pp. 143-211 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Avila, A.; Resende, M. J. Exponential mixing for the Teichmüller flow in the space of quadratic differentials, Comment. Math. Helv., Volume 87 (2012), pp. 589-638 (ISSN: 0010-2571) | DOI | MR | Zbl

Athreya, J. S. Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedicata, Volume 119 (2006), pp. 121-140 (ISSN: 0046-5755) | DOI | MR | Zbl

Bainbridge, M. Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Volume 11 (2007), pp. 1887-2073 (ISSN: 1465-3060) | DOI | MR | Zbl

Bainbridge, M. Billiards in L-shaped tables with barriers, Geom. Funct. Anal., Volume 20 (2010), pp. 299-356 (ISSN: 1016-443X) | DOI | MR | Zbl

Boissy, C. Configurations of saddle connections of quadratic differentials on ℂℙ1 and on hyperelliptic Riemann surfaces, Comment. Math. Helv., Volume 84 (2009), pp. 757-791 (ISSN: 0010-2571) | DOI | MR | Zbl

Calta, K. Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., Volume 17 (2004), pp. 871-908 (ISSN: 0894-0347) | DOI | MR | Zbl

Calta, K.; Wortman, K. On unipotent flows in ℋ(1,1) , Ergodic Theory Dynam. Systems, Volume 30 (2010), pp. 379-398 (ISSN: 0143-3857) | DOI | MR | Zbl

Cohen, M.; Weiss, B. Parking garages with optimal dynamics, Geom. Dedicata, Volume 161 (2012), pp. 157-167 (ISSN: 0046-5755) | DOI | MR | Zbl

Delecroix, V.; Goujard, E.; Zograf, P.; Zorich, A. Integer square-tiled surfaces and interval exchanges of fixed combinatorial type: equidistribution, counting, volumes (in progress)

Douady, A.; Hubbard, J. On the density of Strebel differentials, Invent. math., Volume 30 (1975), pp. 175-179 (ISSN: 0020-9910) | DOI | MR | Zbl

Delecroix, V.; Hubert, P.; Lelièvre, S. Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014), pp. 1085-1110 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Delecroix, V.; Zorich, A. Cries and Whispers in Windtree Forests (preprint arXiv:1502.06405 ) | MR

Eskin, A.; Kontsevich, M.; Zorich, A. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. IHÉS, Volume 120 (2014), pp. 207-333 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Eskin, A.; Mirzakhani, M. Invariant and stationary measures for the SL ( 2 , ℝ ) action on moduli space (preprint arXiv:1302.3320 ) | MR

Eskin, A.; Masur, H. Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, Volume 21 (2001), pp. 443-478 (ISSN: 0143-3857) | DOI | MR | Zbl

Eskin, A.; Mirzakhani, M.; Mohammadi, A. Isolation, equidistribution, and orbit closures for the SL ( 2 , ℝ ) -action on Moduli space (preprint arXiv:1305.3015, to appear in Annals of Math ) | MR

Eskin, A.; Margulis, G.; Mozes, S. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., Volume 147 (1998), pp. 93-141 (ISSN: 0003-486X) | DOI | MR | Zbl

Eskin, A.; Mirzakhani, M.; Rafi, K. Counting closed geodesics in strata (preprint arXiv:1206.5574, to appear in Invent. math ) | MR

Eskin, A.; Masur, H.; Schmoll, M. Billiards in rectangles with barriers, Duke Math. J., Volume 118 (2003), pp. 427-463 (ISSN: 0012-7094) | DOI | MR | Zbl

Eskin, A.; Marklof, J.; Witte Morris, D. Unipotent flows on the space of branched covers of Veech surfaces, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 129-162 (ISSN: 0143-3857) | DOI | MR | Zbl

Eskin, A.; Masur, H.; Zorich, A. Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. IHÉS, Volume 97 (2003), pp. 61-179 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Eskin, A.; Okounkov, A. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. math., Volume 145 (2001), pp. 59-103 (ISSN: 0020-9910) | DOI | MR | Zbl

Eskin, A.; Okounkov, A., Algebraic geometry and number theory (Progr. Math.), Volume 253, Birkhäuser, 2006, pp. 1-25 | DOI | MR | Zbl

Eskin, A.; Okounkov, A.; Pandharipande, R. The theta characteristic of a branched covering, Adv. Math., Volume 217 (2008), pp. 873-888 (ISSN: 0001-8708) | DOI | MR | Zbl

Forni, G. 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) | DOI | MR | Zbl

Gutkin, E.; Judge, C. Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000), pp. 191-213 (ISSN: 0012-7094) | DOI | MR | Zbl

Goujard, E. Volumes of strata of moduli spaces of quadratic differentials: getting explicit values (preprint arXiv:1501.01611 ) | Numdam | MR

Kerckhoff, S.; Masur, H.; Smillie, J. Ergodicity of billiard flows and quadratic differentials, Ann. of Math., Volume 124 (1986), pp. 293-311 (ISSN: 0003-486X) | DOI | MR | Zbl

Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Volume 147 (1992), pp. 1-23 http://projecteuclid.org/euclid.cmp/1104250524 (ISSN: 0010-3616) | DOI | MR | Zbl

Masur, H. Interval exchange transformations and measured foliations, Ann. of Math., Volume 115 (1982), pp. 169-200 (ISSN: 0003-486X) | DOI | MR | Zbl

Masur, H. Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., Volume 53 (1986), pp. 307-314 (ISSN: 0012-7094) | DOI | MR | Zbl

Masur, H., Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) (Math. Sci. Res. Inst. Publ.), Volume 10, Springer, New York, 1988, pp. 215-228 | DOI | MR | Zbl

Masur, H. The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, Volume 10 (1990), pp. 151-176 (ISSN: 0143-3857) | DOI | MR | Zbl

McMullen, C. T. Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 (ISSN: 0894-0347) | DOI | MR | Zbl

McMullen, C. T. Dynamics of SL 2(ℝ) over moduli space in genus two, Ann. of Math., Volume 165 (2007), pp. 397-456 (ISSN: 0003-486X) | DOI | MR | Zbl

Mohanty, S. G. Some convolutions with multinomial coefficients and related probability distributions, SIAM Rev., Volume 8 (1966), pp. 501-509 (ISSN: 0036-1445) | DOI | MR | Zbl

Masur, H.; Smillie, J. Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math., Volume 134 (1991), pp. 455-543 (ISSN: 0003-486X) | DOI | MR | Zbl

Masur, H.; Tabachnikov, S., Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089 | DOI | MR | Zbl

Masur, H.; Zorich, A. Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., Volume 18 (2008), pp. 919-987 (ISSN: 1016-443X) | DOI | MR | Zbl

Nevo, A. Pointwise ergodic theorems for radial averages on simple Lie groups. I, Duke Math. J., Volume 76 (1994), pp. 113-140 (ISSN: 0012-7094) | DOI | MR | Zbl

Nguyen, D.-M. Volumes of the sets of translation surfaces with small saddle connections (preprint arXiv:1211.7314 )

Ríos-Zertuche, R. The pillowcase distribution and near-involutions, Electron. J. Probab., Volume 19 (2014) (ISSN: 1083-6489) | DOI | MR | Zbl

Schwartz, R. E. Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories, Experiment. Math., Volume 18 (2009), pp. 137-171 http://projecteuclid.org/euclid.em/1259158426 (ISSN: 1058-6458) | DOI | MR | Zbl

Veech, W. A. Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., Volume 115 (1982), pp. 201-242 (ISSN: 0003-486X) | DOI | MR | Zbl

Veech, W. A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. math., Volume 97 (1989), pp. 553-583 (ISSN: 0020-9910) | DOI | MR | Zbl

Veech, W. A. Siegel measures, Ann. of Math., Volume 148 (1998), pp. 895-944 (ISSN: 0003-486X) | DOI | MR | Zbl

Vorobets, Y., Algebraic and topological dynamics (Contemp. Math.), Volume 385, Amer. Math. Soc., Providence, RI, 2005, pp. 205-258 | DOI | MR | Zbl

Zorich, A., Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 459-471 | DOI | MR | Zbl

Zorich, A., Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437-583 | DOI | MR | Zbl

Cité par Sources :