# Masur's criterion does not hold in the Thurston metric

## TL;DR

This paper provides a counterexample demonstrating that Masur's criterion does not extend to Teichm"uller space with the Thurston metric, challenging assumptions about geodesic behavior in this setting.

## Contribution

The authors construct a specific counterexample involving a non-uniquely ergodic lamination on a punctured sphere, showing the failure of Masur's criterion in the Thurston metric context.

## Key findings

- Counterexample with a minimal, filling lamination on a seven-times punctured sphere
- Geodesic converging to the lamination remains in the thick part
- Masur's criterion does not hold in the Thurston metric setting

## Abstract

We construct a counterexample for an analogue of Masur's criterion in the setting of Teichm\"uller space equipped with the Thurston metric. For that, we find a minimal, filling, non-uniquely ergodic lamination $\lambda$ on the seven-times punctured sphere with uniformly bounded annular projection distances. Then we show that a geodesic in the corresponding Teichm\"uller space that converges to $\lambda$, stays in the thick part for the whole time.