Limits in 𝒫ℳℱ of Teichmüller geodesics

Abstract

Abstract In this paper we consider the limit set in Thurston’s compactification 𝒫 ⁢ ℳ ⁢ ℱ {\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}} of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.