Siegel–Veech transforms are in <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol{L^2} $\end{document}</tex-math></inline-formula>(with an appendix by Jayadev S. Athreya and Rene Rühr)

Jayadev S. Athreya; Yitwah Cheung; Howard Masur

doi:10.3934/jmd.2019001

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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)

Jayadev S. Athreya ^1, , Yitwah Cheung ^2, and Howard Masur ^3,

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.

Keywords: Siegel–Veech transform, Abelian differentials.

Mathematics Subject Classification: Primary: 32G15; Secondary: 52C23.

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.
[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.
[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.
[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.
[5]	A. Avila, S. 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.
[6]	Y. Cheung, P. 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.
[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.
[8]	B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120. doi: 10.3934/jmd.2019004.
[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.
[10]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[11]	A. Eskin, G. 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.
[12]	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. doi: 10.4007/annals.2015.182.2.7.
[13]	A. Eskin, M. Mirzakhani and K. Rafi, Counting geodesics in a stratum, to appear, Invent. Math.
[14]	S. Fairchild, A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groups, preprint, arXiv: 1901.10115.
[15]	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.
[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.
[17]	M. Magee, R. Rühr and R. Gutiérrez–Romo, Counting saddle connections in a homology class modulo $\mathcal q$, preprint, arXiv: 1809.00579, 2018.
[18]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459.
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[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.
[21]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, preprint, arXiv: 1708.06263.
[22]	D. Nguyen, Volume of the set of surfaces with small saddle connection in rank one affine manifolds, preprint, arXiv: 1211.7314.
[23]	K. Rafi, Hyperbolicity in Teichmüller space, Geometry Topology, 18 (2014), 3025-3053. doi: 10.2140/gt.2014.18.3025.
[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.
[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.
[26]	C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347. doi: 10.2307/1969027.
[27]	J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557. doi: 10.1007/s00222-010-0236-0.
[28]	W. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033.
[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.
[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.
[31]	A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geometry Topology, 19 (2015), 413-438. doi: 10.2140/gt.2015.19.413.
[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.

show all references

References:

[1]	J. S. Athreya, Random affine lattices, Contemp. Math., 639 (2015), 160-174. doi: 10.1090/conm/639/12793.
[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.
[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.
[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.
[5]	A. Avila, S. 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.
[6]	Y. Cheung, P. 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.
[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.
[8]	B. Dozier, Equidistribution of saddle connections on translation surfaces, J. Mod. Dyn., 14 (2019), 87-120. doi: 10.3934/jmd.2019004.
[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.
[10]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[11]	A. Eskin, G. 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.
[12]	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. doi: 10.4007/annals.2015.182.2.7.
[13]	A. Eskin, M. Mirzakhani and K. Rafi, Counting geodesics in a stratum, to appear, Invent. Math.
[14]	S. Fairchild, A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groups, preprint, arXiv: 1901.10115.
[15]	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.
[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.
[17]	M. Magee, R. Rühr and R. Gutiérrez–Romo, Counting saddle connections in a homology class modulo $\mathcal q$, preprint, arXiv: 1809.00579, 2018.
[18]	H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems, 10 (1990), 151-176. doi: 10.1017/S0143385700005459.
[19]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[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.
[21]	A. Nevo, R. Rühr and B. Weiss, Effective counting on translation surfaces, preprint, arXiv: 1708.06263.
[22]	D. Nguyen, Volume of the set of surfaces with small saddle connection in rank one affine manifolds, preprint, arXiv: 1211.7314.
[23]	K. Rafi, Hyperbolicity in Teichmüller space, Geometry Topology, 18 (2014), 3025-3053. doi: 10.2140/gt.2014.18.3025.
[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.
[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.
[26]	C. L. Siegel, A mean value theorem in geometry of numbers, Ann. Math., 46 (1945), 340-347. doi: 10.2307/1969027.
[27]	J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557. doi: 10.1007/s00222-010-0236-0.
[28]	W. Veech, Siegel measures, Ann. of Math. (2), 148 (1998), 895-944. doi: 10.2307/121033.
[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.
[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.
[31]	A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geometry Topology, 19 (2015), 413-438. doi: 10.2140/gt.2015.19.413.
[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.

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

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

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