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

Jayadev S. Athreya; Yitwah Cheung; Howard Masur

doi:10.3934/jmd.2019001

Article Contents

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

This volume Previous Article Bill Veech's contributions to dynamical systems Next Article The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

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

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

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Abstract

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.

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References

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