Convergence of Siegel–Veech constants

Abstract

We show that for any weakly convergent sequence of ergodic SL2(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.

Equidistribution of saddle connections on translation surfaces

“…The new elements here are (i) integration over a fixed interval of angles, rather than the whole interval or a changing interval, and (ii) the resulting additive term b · |I|. This proposition is also the key tool needed to prove a result on convergence of Siegel-Veech constants in [Doz18]; for that application it is essential that the constant b does not depend on the surface X. For all applications, it is crucial that the function α does not depend on T .…”

Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)

“…which gives (2) for every flat surface as R → ∞ outside a zero upper logarithmic density set of radii that depends on the flat surface. The best known results for asymptotic counting that hold on all surfaces are Eskin, Mirzakhani, and Mohammadi's result [8,Theorem 2.12] that every surface has an asymptotic growth of saddle connections on average, and Dozier's result [4] building on techniques in [7] that the constants c 1 , c 2 in (1) can be chosen to depend only on the connected component of the stratum in question. The best almost everywhere counting result is the following theorem of Nevo, Rühr, and Weiss.…”