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 ${GL}^{+}\left(2,ℝ\right)$–orbit closure of $M$. Applications are given concerning complete periodicity, affine field of definition and the number of parallel cylinders which may be found on a translation surface in a given orbit closure.