On the classification of mapping class actions on Thurstons asymmetric metric

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmuller space of that surface equipped with Thurstons 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 Thurstons classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmuller space equipped with Teichmullers metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmuller space equipped with the Weil-Petersson metric.