Convergence of Siegel-Veech constants

TL;DR

This paper proves the convergence of Siegel-Veech constants for sequences of ergodic measures on translation surfaces and applies this to confirm a conjecture about Teichmüller curves in genus two, also providing bounds on saddle connections.

Contribution

It establishes the convergence of Siegel-Veech constants for weakly convergent ergodic measures and confirms a conjecture for genus two Teichmüller curves, using recurrence techniques.

Findings

Siegel-Veech constants converge under weak measure convergence

Confirmed convergence for sequences of Teichmüller curves in genus two

Provided uniform quadratic bounds on saddle connections

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.