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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 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.

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          Affine mappings of translation surfaces: geometry and arithmetic

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            Asymptotic formulas on flat surfaces

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              Billiards in rectangles with barriers

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                Author and article information

                Journal
                2016-12-31
                Article
                1701.00175
                04aa9f8c-b072-4645-953e-e6ca910c7370

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                19 pages, 1 figure
                math.DS math.GT

                Differential equations & Dynamical systems,Geometry & Topology
                Differential equations & Dynamical systems, Geometry & Topology

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