On the classification of mapping class actions on Thurston's asymmetric metric

TL;DR

This paper classifies the actions of the mapping class group on Teichmüller space with Thurston's asymmetric metric into elliptic, parabolic, hyperbolic, and pseudo-hyperbolic types, relating them to Thurston's classification.

Contribution

It provides a new classification framework for mapping class actions on Teichmüller space with Thurston's metric, extending previous work on other metrics.

Findings

Classified actions as elliptic, parabolic, hyperbolic, or pseudo-hyperbolic.

Connected these classifications to Thurston's classification of mapping classes.

Extended the understanding of mapping class actions in the context of Thurston's asymmetric metric.

Abstract

We study the action of the elements of the mapping class group of a surface of finite type on the Teichm\"uller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichm\"uller space equipped with Teichm\"uller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichm\"uller space equipped with the Weil-Petersson metric.