Convergence of Siegel–Veech constants

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 Teichmuller 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.