Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

Michael Magee; Rene Rühr

doi:10.3934/jmd.2019020

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2019, Volume 15: 237-262. Doi: 10.3934/jmd.2019020

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Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo)

Michael Magee^1, and
Rene Rühr^2,

1.
Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Rd, Durham DH1 3LE, United Kingdom
2.
Faculty of Mathematics, Technion, Haifa, 32000 Israel

(with an appendix by Rodolfo Gutiérrez-Romo)
Rodolfo Gutiérrez-Romo < rodolfo.gutierrez@imj-prg.fr>: Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586, Bátiment Sophie Germain, 75205 PARIS Cedex 13, France

Received: November 29, 2018

Revised: May 09, 2019

Published: August 2019

Supported in part by the S.N.F., project number 168823

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Abstract

We give effective estimates for the number of saddle connections on a translation surface that have length $ \leq L $ and are in a prescribed homology class modulo $ q $. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.

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Mathematics Subject Classification: Primary: 32G15, 11N45; Secondary: 37E35, 20H05, 37A30.

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