On continuity of drifts of the mapping class group

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the “translation distance” of the random walk. In this paper, we prove that the drift varies continuously with the transition probability measures, under the assumption that the distance and the horofunctions on $X$ are expressed by certain ratios. As an example, we consider the mapping class group MCG($S$) acting on the Teichmüller space. By using north-south dynamics, we also consider the continuity of the drift for a sequence converging to a Dirac measure. As an appendix, we prove that the asymptotic entropy of the random walks on MCG($S$) varies continuously.

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This work was partially supported by JSPS KAKENHI Grant Number 19K14525.

Received 19 August 2019

Accepted 27 January 2020

Published 2 June 2021