Measure bound for translation surfaces with short saddle connections
Abstract
We prove that any ergodic $SL_2(R)$invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two nonparallel saddle connections of length at most $\epsilon_1, \epsilon_2$ is $O(\epsilon_1^2 \epsilon_2^2)$. We prove a more general theorem which works for any number of short saddle connections. The proof uses the multiscale compactification of strata recently introduced by BainbridgeChenGendronGrushevskyMöller and the algebraicity result of Filip.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.10026
 arXiv:
 arXiv:2002.10026
 Bibcode:
 2020arXiv200210026D
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Algebraic Geometry;
 Mathematics  Geometric Topology
 EPrint:
 57 pages, 9 figures. Added Remark 1.5 concerning the opposite inequality. Various minor changes throughout. To appear in GAFA