# American Institute of Mathematical Sciences

2018, 12: 55-122. doi: 10.3934/jmd.2018004

## Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland 2 Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, PR China 3 School of Mathematics, University of Bristol, Howard House, Queens Ave, BS8 1SD, Bristol, United Kingdom

Received  June 13, 2016 Revised  November 06, 2017 Published  March 2018

Fund Project: KF: Research partially supported by the National Science Centre (Poland) grant 2014/13/ B/ST1/03153.
RS: Supported by NSFC (11201388), NSFC (11271278), ERC starter grant DLGAPS 279893.
CU: Supported by the ERC, via the Starting Grant ChaParDyn, as well as by the Leverhulme Trust via a Leverhulme Prize and by the Royal Society and the Wolfson Foundation via a Royal Society Wolfson Research Merit Award. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 335989.

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices $ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathscr{H}(1,1)$ of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

Citation: Krzysztof Frączek, Ronggang Shi, Corinna Ulcigrai. Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. Journal of Modern Dynamics, 2018, 12: 55-122. doi: 10.3934/jmd.2018004
##### References:
 [1] A. Avila, A. Eskin and M. Möller, Symplectic and isometric SL(2, R)-invariant subbundles of the Hodge bundle, J. Reine Angew. Math., 732 (2017), 1-20. Google Scholar [2] A. Avila, S. Gouëzel and J.-Ch. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. Google Scholar [3] A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2018). inéaire, (2018). doi: 10.1016/j.anihpc.2017.11.006. Google Scholar [4] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887. Google Scholar [5] T. Browning and I. Vinogradov, Effective Ratner theorem for $SL(2, \mathbb R) <\ltimes \mathbb R^2$ and gaps in $\sqrt n$ modulo $1$, J. Lond. Math. oc. (2), 94 (2016), 61-84. doi: 10.1112/jlms/jdw025. Google Scholar [6] Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅱ), J. Amer. Math. Soc., 26 (2013), 659-734. doi: 10.1090/S0894-0347-2013-00760-2. Google Scholar [7] J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23. doi: 10.3934/jmd.2015.9.1. Google Scholar [8] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110. doi: 10.24033/asens.2234. Google Scholar [9] V. Dragović and M. Radnović, Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn., 8 (2014), 109-132. doi: 10.3934/jmd.2014.8.109. Google Scholar [10] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010,155–241. Google Scholar [11] N. D. Elkies and C. T. McMullen, Gaps in $\sqrt{n}$ mod $1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139. doi: 10.1215/S0012-7094-04-12314-0. Google Scholar [12] A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, arXiv: 1702.02074.Google Scholar [13] A. Eskin, G. A. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar [14] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL_2(\mathbb{R})$ action on moduli space, arXiv: 1302.3320.Google Scholar [15] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution and orbit closures for the $SL(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721. Google Scholar [16] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670. doi: 10.1007/s00222-015-0643-3. Google Scholar [17] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. Google Scholar [18] R. Fox and R. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150. doi: 10.1215/S0012-7094-36-00213-2. Google Scholar [19] K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., to appear, arXiv: 1506.05884.Google Scholar [20] K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses, Nonlinearity, 27 (2014), 1689-1707. doi: 10.1088/0951-7715/27/7/1689. Google Scholar [21] K. Frączek and C. Ulcigrai, Non-ergodic $\mathbb{Z}$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z. Google Scholar [22] I. Ya. Goldsheid and G. A. Margulis, Lyapunov exponents of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71. Google Scholar [23] Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802. doi: 10.1017/S0143385703000440. Google Scholar [24] J. H. Hannay and T. M. Haeusserab, Retroreflection by refraction, J. Mod. Opt., 40 (1993), 1437-1442. doi: 10.1080/09500349314551501. Google Scholar [25] A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian), Mat.Zametki, 18 (1975), 291-300. Google Scholar [26] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280. Google Scholar [27] D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997. Google Scholar [28] D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879. doi: 10.1007/s00208-016-1404-3. Google Scholar [29] R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816. Google Scholar [30] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387-442. Google Scholar [31] Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math., 552 (2002), 131-177. Google Scholar [32] Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 245-284. doi: 10.24033/asens.2214. Google Scholar [33] S. Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory Dynam. Systems, 15 (1995), 1207-1210. doi: 10.1017/S0143385700009871. Google Scholar [34] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357. Google Scholar [35] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 275-306. Google Scholar [36] N. Shah, Equidistribution of expanding translates of curves and Dirichlet's theorem on diophantine approximation, Invent. Math, 177 (2009), 509-532. doi: 10.1007/s00222-009-0186-6. Google Scholar [37] N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279. doi: 10.1215/00127094-2009-026. Google Scholar [38] R. Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space, arXiv: 1405.2067.Google Scholar [39] R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv: 1510.05256.Google Scholar [40] A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000. Google Scholar [41] S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI, 2005. Google Scholar [42] A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215. Google Scholar [43] A. Zorich, How do the leaves of a closed 1-form wind around a surface?, Amer. Math. Soc. Transl. Ser. 2, 197 (1999), 135-178. Google Scholar

show all references

##### References:
 [1] A. Avila, A. Eskin and M. Möller, Symplectic and isometric SL(2, R)-invariant subbundles of the Hodge bundle, J. Reine Angew. Math., 732 (2017), 1-20. Google Scholar [2] A. Avila, S. Gouëzel and J.-Ch. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. Google Scholar [3] A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2018). inéaire, (2018). doi: 10.1016/j.anihpc.2017.11.006. Google Scholar [4] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887. Google Scholar [5] T. Browning and I. Vinogradov, Effective Ratner theorem for $SL(2, \mathbb R) <\ltimes \mathbb R^2$ and gaps in $\sqrt n$ modulo $1$, J. Lond. Math. oc. (2), 94 (2016), 61-84. doi: 10.1112/jlms/jdw025. Google Scholar [6] Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅱ), J. Amer. Math. Soc., 26 (2013), 659-734. doi: 10.1090/S0894-0347-2013-00760-2. Google Scholar [7] J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23. doi: 10.3934/jmd.2015.9.1. Google Scholar [8] V. Delecroix, P. Hubert and S. Lelièvre, Diffusion for the periodic wind-tree model, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 1085-1110. doi: 10.24033/asens.2234. Google Scholar [9] V. Dragović and M. Radnović, Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn., 8 (2014), 109-132. doi: 10.3934/jmd.2014.8.109. Google Scholar [10] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010,155–241. Google Scholar [11] N. D. Elkies and C. T. McMullen, Gaps in $\sqrt{n}$ mod $1$ and ergodic theory, Duke Math. J., 123 (2004), 95-139. doi: 10.1215/S0012-7094-04-12314-0. Google Scholar [12] A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, arXiv: 1702.02074.Google Scholar [13] A. Eskin, G. A. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar [14] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $SL_2(\mathbb{R})$ action on moduli space, arXiv: 1302.3320.Google Scholar [15] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution and orbit closures for the $SL(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721. Google Scholar [16] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., 205 (2016), 617-670. doi: 10.1007/s00222-015-0643-3. Google Scholar [17] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150. Google Scholar [18] R. Fox and R. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Math. J., 2 (1936), 147-150. doi: 10.1215/S0012-7094-36-00213-2. Google Scholar [19] K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., to appear, arXiv: 1506.05884.Google Scholar [20] K. Frączek and M. Schmoll, Directional localization of light rays in a periodic array of retro-reflector lenses, Nonlinearity, 27 (2014), 1689-1707. doi: 10.1088/0951-7715/27/7/1689. Google Scholar [21] K. Frączek and C. Ulcigrai, Non-ergodic $\mathbb{Z}$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298. doi: 10.1007/s00222-013-0482-z. Google Scholar [22] I. Ya. Goldsheid and G. A. Margulis, Lyapunov exponents of a product of random matrices, Russian Math. Surveys, 44 (1989), 11-71. Google Scholar [23] Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802. doi: 10.1017/S0143385703000440. Google Scholar [24] J. H. Hannay and T. M. Haeusserab, Retroreflection by refraction, J. Mod. Opt., 40 (1993), 1437-1442. doi: 10.1080/09500349314551501. Google Scholar [25] A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons, (Russian), Mat.Zametki, 18 (1975), 291-300. Google Scholar [26] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2), 124 (1986), 293-311. doi: 10.2307/1971280. Google Scholar [27] D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997. Google Scholar [28] D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann., 367 (2017), 857-879. doi: 10.1007/s00208-016-1404-3. Google Scholar [29] R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816. Google Scholar [30] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J., 66 (1992), 387-442. Google Scholar [31] Y. Minsky and B. Weiss, Nondivergence of horocyclic flows on moduli space, J. Reine Angew. Math., 552 (2002), 131-177. Google Scholar [32] Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 245-284. doi: 10.24033/asens.2214. Google Scholar [33] S. Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory Dynam. Systems, 15 (1995), 1207-1210. doi: 10.1017/S0143385700009871. Google Scholar [34] M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607. doi: 10.2307/2944357. Google Scholar [35] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 275-306. Google Scholar [36] N. Shah, Equidistribution of expanding translates of curves and Dirichlet's theorem on diophantine approximation, Invent. Math, 177 (2009), 509-532. doi: 10.1007/s00222-009-0186-6. Google Scholar [37] N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279. doi: 10.1215/00127094-2009-026. Google Scholar [38] R. Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space, arXiv: 1405.2067.Google Scholar [39] R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv: 1510.05256.Google Scholar [40] A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000. Google Scholar [41] S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, 30, American Mathematical Society, Providence, RI, 2005. Google Scholar [42] A. Zorich, Deviation for interval exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215. Google Scholar [43] A. Zorich, How do the leaves of a closed 1-form wind around a surface?, Amer. Math. Soc. Transl. Ser. 2, 197 (1999), 135-178. Google Scholar
The table $\mathscr{D}_{\lambda_0}$ and two types of invariant sets for the billiard flow
Eaton lens and a parallel family of light rays
The system of lenses $L(\Lambda,R)$
A translation surface $(M, \omega)$ in the space $\mathscr{M}^{dc}$ of double covers of tori.
The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $\lambda_0 <\lambda< b$
The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $b<\lambda< a$
The surface $({M}_\lambda, \omega_\lambda)$
The system of lenses $F(\Lambda,R,\theta)$
The surface $M(\Lambda,R)$
The triangle $\triangle ABC$ and the line $x = 2sy+s^2$
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