Cylinder deformations in orbit closures of translation surfaces

TL;DR

This paper investigates how specific cylinder deformations in translation surfaces influence their orbit closures under GL(2,R), with applications to periodicity, field of definition, and cylinder counts, using recent orbit closure theorems.

Contribution

It introduces new results on cylinder deformations within orbit closures, extending understanding of translation surface dynamics and their geometric properties.

Findings

Certain cylinder deformations stay within the orbit closure.

Results on complete periodicity and field of definition.

Bounds on the number of parallel cylinders in orbit closures.

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.