• DOI:10.1007/s10711-018-0332-7
• Corpus ID: 119174914

Convergence of Siegel–Veech constants

@article{Dozier2016ConvergenceOS,
  title={Convergence of Siegel–Veech constants},
  author={Benjamin Dozier},
  journal={Geometriae Dedicata},
  year={2016},
  volume={198},
  pages={131-142}
}

• B. Dozier
• Published 31 December 2016
• Mathematics
• Geometriae Dedicata

We show that for any weakly convergent sequence of ergodic $$SL_2(\mathbb {R})$$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.