Cylinder deformations in orbit closures of translation surfaces

Abstract

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.

v2: Minor revision. Comments welcome! 24 pages

References in corpus (1)

• The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

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