Convergence of Siegel-Veech constants
| Autor: | Dozier, Benjamin |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 Teichm\"uller 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. Comment: 12 pages; replaced proof of the key technical tool Proposition 1.1 with a reference to a more general result proved by the author in arXiv:1705.10847; other minor changes |
| Databáze: | arXiv |
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