Weak mixing directions in non-arithmetic Veech surfaces

Avila, Artur; Delecroix, Vincent

doi:10.1090/jams/856

Weak mixing directions in non-arithmetic Veech surfaces
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by Artur Avila and Vincent Delecroix

J. Amer. Math. Soc. 29 (2016), 1167-1208

DOI: https://doi.org/10.1090/jams/856

Published electronically: January 13, 2016

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Abstract:

We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square, and hexagon). More generally, we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non-arithmetic Veech surface and show that the Hausdorff dimension of the set of non-weak mixing directions is not full. We also provide a necessary condition, verified, for instance, by the Veech surface corresponding to the billiard in the pentagon, for the set of non-weak mixing directions to have a positive Hausdorff dimension.

References

Artur Avila and Vincent Delecroix, Large deviations for algebraic $\textrm {SL}_2(\mathbb {R})$-invariant measures in moduli space. In progress.
Artur Avila and Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2) 165 (2007), no. 2, 637–664. MR 2299743, DOI 10.4007/annals.2007.165.637
Artur Avila and Giovanni Forni, Weak mixing in L shaped billiards. In progress.
Artur Avila and Marcelo Viana, Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Math. 198 (2007), no. 1, 1–56. MR 2316268, DOI 10.1007/s11511-007-0012-1
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Irene I. Bouw and Martin Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), no. 1, 139–185. MR 2680418, DOI 10.4007/annals.2010.172.139
Xavier Bressaud, Alexander I. Bufetov, and Pascal Hubert, Deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 483–522. MR 3254932, DOI 10.1112/plms/pdu009
Xavier Bressaud, Fabien Durand, and Alejandro Maass, Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems, J. London Math. Soc. (2) 72 (2005), no. 3, 799–816. MR 2190338, DOI 10.1112/S0024610705006800
Xavier Bressaud, Fabien Durand, and Alejandro Maass, On the eigenvalues of finite rank Bratteli-Vershik dynamical systems, Ergodic Theory Dynam. Systems 30 (2010), no. 3, 639–664. MR 2643706, DOI 10.1017/S0143385709000236
Kariane Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004), no. 4, 871–908. MR 2083470, DOI 10.1090/S0894-0347-04-00461-8
Jon Chaika and Alex Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn. 9 (2015), 1–23. MR 3395258, DOI 10.3934/jmd.2015.9.1
Kariane Calta and John Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn. 2 (2008), no. 2, 209–248. MR 2383267, DOI 10.3934/jmd.2008.2.209
Alex Eskin and Carlos Matheus, A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves, Geom. Dedicata 179 (2015), 45–67. MR 3424657, DOI 10.1007/s10711-015-0067-7
Giovanni Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103. MR 1888794, DOI 10.2307/3062150
M. Guenais and F. Parreau, Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier, available at arXiv:math/0605250v1.
Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213. MR 1760625, DOI 10.1215/S0012-7094-00-10321-3
W. Patrick Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. IMRN 12 (2013), 2657–2698. MR 3071661, DOI 10.1093/imrn/rns124
Pascal Hubert, Luca Marchese, and Corinna Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal. 25 (2015), no. 1, 180–255. MR 3320892, DOI 10.1007/s00039-015-0321-z
Pascal Hubert and Thomas A. Schmidt, An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 501–526. MR 2186246, DOI 10.1016/S1874-575X(06)80031-7
Anatole Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math. 35 (1980), no. 4, 301–310. MR 594335, DOI 10.1007/BF02760655
Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31. MR 357739, DOI 10.1007/BF01236981
Richard Kenyon and John Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000), no. 1, 65–108. MR 1760496, DOI 10.1007/s000140050113
Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293–311. MR 855297, DOI 10.2307/1971280
Pascal Hubert and Erwan Lanneau, Veech groups without parabolic elements, Duke Math. J. 133 (2006), no. 2, 335–346. MR 2225696, DOI 10.1215/S0012-7094-06-13326-4
Erwan Lanneau and Duc-Manh Nguyen, Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4, J. Topol. 7 (2014), no. 2, 475–522. MR 3217628, DOI 10.1112/jtopol/jtt036
Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
Howard Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442. MR 1167101, DOI 10.1215/S0012-7094-92-06613-0
Curtis T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885. MR 1992827, DOI 10.1090/S0894-0347-03-00432-6
Curtis T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006), no. 3, 569–590. MR 2228463, DOI 10.1215/S0012-7094-06-13335-5
Curtis T. McMullen, Dynamics of $\textrm {SL}_2(\Bbb R)$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), no. 2, 397–456. MR 2299738, DOI 10.4007/annals.2007.165.397
Stefano Marmi, Pierre Moussa, and Jean-Cristophe Yoccoz, The cohomological equation for Roth type interval exchange transformations, J. Amer. Math. Soc. 18 (2005), 823–872.
Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015–1089. MR 1928530, DOI 10.1016/S1874-575X(02)80015-7
Martin Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327–344. MR 2188128, DOI 10.1090/S0894-0347-05-00512-6
William A. Stein et al., Sage Mathematics Software (Version 5.0). The Sage Development Team, http://www.sagemath.org.
John Smillie and Barak Weiss, Characterizations of lattice surfaces, Invent. Math. 180 (2010), no. 3, 535–557. MR 2609249, DOI 10.1007/s00222-010-0236-0
William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
William A. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math. 106 (1984), no. 6, 1331–1359. MR 765582, DOI 10.2307/2374396
W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006, DOI 10.1007/BF01388890
Clayton C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 1019–1042. MR 1645350, DOI 10.1017/S0143385798117479
Alex Wright, Schwarz triangle mappings and Teichmüller curves: the Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal. 23 (2013), no. 2, 776–809. MR 3053761, DOI 10.1007/s00039-013-0221-z