2019, 14: 1-19. doi: 10.3934/jmd.2019001

Siegel–Veech transforms are in $ \boldsymbol{L^2} $(with an appendix by Jayadev S. Athreya and Rene Rühr)

1. 

Department of Mathematics, University of Washington, Padelford Hall, Seattle, WA 98195, USA

2. 

Department of Mathematics, San Francisco State University, Thornton Hall 937, 1600 Holloway Ave, San Francisco, CA 94132, USA

3. 

Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60615, USA

4. 

Faculty of Mathematics, Technion, Haifa, 32000 Israel

Dedicated to the memory of William Veech

Received  November 29, 2017 Revised  February 12, 2019 Published  March 2019

Fund Project: JSA: Partially supported by NSF CAREER grant DMS 1559860.
YC: Partially supported by NSF DMS 1600476.
HM: Partially supported by NSF DMS 1607512

Let $\mathscr{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on $\mathbb{R}^2$ is in $L^2(\mathscr{H}, \mu)$, where $\mu$ is the Lebesgue measure on $\mathscr{H}$, and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to $SL(2,\mathbb{R})$-invariant measures on strata satisfying certain integrability conditions.

Citation: Jayadev S. Athreya, Yitwah Cheung, Howard Masur. Siegel–Veech transforms are in $ \boldsymbol{L^2} $(with an appendix by Jayadev S. Athreya and Rene Rühr). Journal of Modern Dynamics, 2019, 14: 1-19. doi: 10.3934/jmd.2019001
References:
[1]

J. S. Athreya, Random affine lattices, Contemp. Math., 639 (2015), 160-174. doi: 10.1090/conm/639/12793. Google Scholar

[2]

J. S. Athreya and J. Chaika, The distribution of gaps for saddle connection directions, Geom. Funct. Anal., 22 (2012), 1491-1516. doi: 10.1007/s00039-012-0164-9. Google Scholar

[3]

J. S. Athreya and G. A. Margulis, Logarithm laws for unipotent flows, I, J. Mod. Dyn., 3 (2009), 359-378. doi: 10.3934/jmd.2009.3.359. Google Scholar

[4]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn., 12 (2018), 9-16. doi: 10.3934/jmd.2018002. Google Scholar

[5]

A. AvilaS. Gouëzel and J.-C. Yoccoz, Exponential mixing for the {T}eichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. Google Scholar

[6]

Y. CheungP. Hubert and H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. Math., 183 (2011), 337-383. doi: 10.1007/s00222-010-0279-2. Google Scholar

[7]

P. Chew, There is a planar graph almost as good as the complete graph, SCG '86 Proceedings of the Second Annual Symposium on Computational Geometry, 1986,169–177. doi: 10.1145/10515.10534. Google Scholar

[8]

B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120. doi: 10.3934/jmd.2019004. Google Scholar

[9]

A. Eskin, Counting problems in moduli space, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,581–595. doi: 10.1016/S1874-575X(06)80034-2. Google Scholar

[10]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. Google Scholar

[11]

A. EskinG. 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

[12]

A. EskinM. 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. doi: 10.4007/annals.2015.182.2.7. Google Scholar

[13]

A. Eskin, M. Mirzakhani and K. Rafi, Counting geodesics in a stratum, to appear, Invent. Math.Google Scholar

[14]

S. Fairchild, A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groups, preprint, arXiv: 1901.10115.Google Scholar

[15]

S. KerckhoffH. 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

[16]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x. Google Scholar

[17]

M. Magee, R. Rühr and R. Gutiérrez–Romo, Counting saddle connections in a homology class modulo $\mathcal q$, preprint, arXiv: 1809.00579, 2018.Google Scholar

[18]

H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459. Google Scholar

[19]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar

[20]

H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2), 134 (1991), 455-543. doi: 10.2307/2944356. Google Scholar

[21]

A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, preprint, arXiv: 1708.06263.Google Scholar

[22]

D. Nguyen, Volume of the set of surfaces with small saddle connection in rank one affine manifolds, preprint, arXiv: 1211.7314.Google Scholar

[23]

K. Rafi, Hyperbolicity in Teichmüller space, Geometry Topology, 18 (2014), 3025-3053. doi: 10.2140/gt.2014.18.3025. Google Scholar

[24]

C. A. Rogers, The number of lattice points in a set, Proc. London Math. Soc. (3), 6 (1956), 305-320. doi: 10.1112/plms/s3-6.2.305. Google Scholar

[25]

W. Schmidt, A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529. doi: 10.1090/S0002-9947-1960-0117222-9. Google Scholar

[26]

C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347. doi: 10.2307/1969027. Google Scholar

[27]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557. doi: 10.1007/s00222-010-0236-0. Google Scholar

[28]

W. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033. Google Scholar

[29]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. Google Scholar

[30]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. Google Scholar

[31]

A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geometry Topology, 19 (2015), 413-438. doi: 10.2140/gt.2015.19.413. Google Scholar

[32]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,437–583. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

show all references

References:
[1]

J. S. Athreya, Random affine lattices, Contemp. Math., 639 (2015), 160-174. doi: 10.1090/conm/639/12793. Google Scholar

[2]

J. S. Athreya and J. Chaika, The distribution of gaps for saddle connection directions, Geom. Funct. Anal., 22 (2012), 1491-1516. doi: 10.1007/s00039-012-0164-9. Google Scholar

[3]

J. S. Athreya and G. A. Margulis, Logarithm laws for unipotent flows, I, J. Mod. Dyn., 3 (2009), 359-378. doi: 10.3934/jmd.2009.3.359. Google Scholar

[4]

J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn., 12 (2018), 9-16. doi: 10.3934/jmd.2018002. Google Scholar

[5]

A. AvilaS. Gouëzel and J.-C. Yoccoz, Exponential mixing for the {T}eichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5. Google Scholar

[6]

Y. CheungP. Hubert and H. Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. Math., 183 (2011), 337-383. doi: 10.1007/s00222-010-0279-2. Google Scholar

[7]

P. Chew, There is a planar graph almost as good as the complete graph, SCG '86 Proceedings of the Second Annual Symposium on Computational Geometry, 1986,169–177. doi: 10.1145/10515.10534. Google Scholar

[8]

B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120. doi: 10.3934/jmd.2019004. Google Scholar

[9]

A. Eskin, Counting problems in moduli space, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,581–595. doi: 10.1016/S1874-575X(06)80034-2. Google Scholar

[10]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225. Google Scholar

[11]

A. EskinG. 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

[12]

A. EskinM. 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. doi: 10.4007/annals.2015.182.2.7. Google Scholar

[13]

A. Eskin, M. Mirzakhani and K. Rafi, Counting geodesics in a stratum, to appear, Invent. Math.Google Scholar

[14]

S. Fairchild, A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groups, preprint, arXiv: 1901.10115.Google Scholar

[15]

S. KerckhoffH. 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

[16]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x. Google Scholar

[17]

M. Magee, R. Rühr and R. Gutiérrez–Romo, Counting saddle connections in a homology class modulo $\mathcal q$, preprint, arXiv: 1809.00579, 2018.Google Scholar

[18]

H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459. Google Scholar

[19]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341. Google Scholar

[20]

H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. (2), 134 (1991), 455-543. doi: 10.2307/2944356. Google Scholar

[21]

A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, preprint, arXiv: 1708.06263.Google Scholar

[22]

D. Nguyen, Volume of the set of surfaces with small saddle connection in rank one affine manifolds, preprint, arXiv: 1211.7314.Google Scholar

[23]

K. Rafi, Hyperbolicity in Teichmüller space, Geometry Topology, 18 (2014), 3025-3053. doi: 10.2140/gt.2014.18.3025. Google Scholar

[24]

C. A. Rogers, The number of lattice points in a set, Proc. London Math. Soc. (3), 6 (1956), 305-320. doi: 10.1112/plms/s3-6.2.305. Google Scholar

[25]

W. Schmidt, A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529. doi: 10.1090/S0002-9947-1960-0117222-9. Google Scholar

[26]

C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347. doi: 10.2307/1969027. Google Scholar

[27]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557. doi: 10.1007/s00222-010-0236-0. Google Scholar

[28]

W. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033. Google Scholar

[29]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. Google Scholar

[30]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391. Google Scholar

[31]

A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geometry Topology, 19 (2015), 413-438. doi: 10.2140/gt.2015.19.413. Google Scholar

[32]

A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006,437–583. doi: 10.1007/978-3-540-31347-2_13. Google Scholar

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