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Convergence of Siegel–Veech constants

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Abstract

We show that for any weakly convergent sequence of ergodic \(SL_2(\mathbb {R})\)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel–Veech constants converge to the Siegel–Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin–Mirzakhani–Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel–Veech constants associated to Teichmüller curves in genus two. The proof uses a recurrence result closely related to techniques developed by Eskin–Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface.

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Acknowledgements

I would like to thank Maryam Mirzakhani, my thesis advisor, for guiding me with numerous stimulating conversations and suggestions. I am also very grateful to Alex Wright, for many helpful discussions and detailed feedback.

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  1. Department of Mathematics, Stanford University, Stanford, CA, USA

    Benjamin Dozier

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  1. Benjamin Dozier
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Correspondence to Benjamin Dozier.

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Supported in part by NSF Grant DGE-114747.

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Dozier, B. Convergence of Siegel–Veech constants. Geom Dedicata 198, 131–142 (2019). https://doi.org/10.1007/s10711-018-0332-7

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  • DOI: https://doi.org/10.1007/s10711-018-0332-7

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