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arXiv · 1701.00175

Convergence of Siegel-Veech constants

Benjamin Dozier
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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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Benjamin Dozier. 2017-10-30. Convergence of Siegel-Veech constants. https://doi.org/10.1007/s10711-018-0332-7

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