A Full Study of the Dynamics on One-Holed Dilation Tori

Abstract

An open question in the study of dilation surfaces is to determine the typical dynamical behavior of the directional flow on a fixed dilation surface. We show that on any one-holed dilation torus, in all but a measure zero Cantor set of directions, the directional flow has an attracting periodic orbit, is minimal, or is completely periodic. We further show that for directions in this Cantor set, the directional flow is attracted to either a saddle connection or a lamination on the surface that is locally the product of a measure zero Cantor set and an interval.