Limit sets of Weil-Petersson geodesics

TL;DR

This paper studies the limit sets of Weil-Petersson geodesic rays, proving they are single points for uniquely ergodic laminations and circles for non-uniquely ergodic laminations, revealing diverse boundary behaviors.

Contribution

It establishes the precise nature of limit sets for Weil-Petersson geodesics based on ergodic properties of ending laminations, including new examples with complex limit sets.

Findings

Limit set is a single point for uniquely ergodic laminations.

Limit set is a circle for non-uniquely ergodic laminations.

Provides explicit examples of geodesics with non-trivial limit sets.

Abstract

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichm\"uller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification.