Masur’s criterion does not hold in the Thurston metric

Abstract

We show that there is a minimal, filling, and non-uniquely ergodic lamination $\lambda$ on the seven-times punctured sphere with uniformly bounded annular projection distances. Moreover, we show that there is a geodesic in the thick part of the corresponding Teichmüller space equipped with the Thurston metric which converges to $\lambda$. This provides a counterexample to an analog of Masur's criterion for Teichmüller space equipped with the Thurston metric.