Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation

Abstract

We describe a method for constructing Teichmüller geodesics where the vertical measured foliation $ν$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the set of all possible measured foliations in the homotopy class of $ν$. With further adjustment of the parameters, one can even take $ν$ to be an ergodic measure on a non-uniquely ergodic foliation.

v1: 28 pages, 4 figures. v2: Fixed figures and minor typos, added references. v3: Final version accepted for publication. Includes a relaxed hypothesis in Theorem 1.1 and improvements to exposition following referee's comments. v4: Updated references