On continuity of drifts of the mapping class group

TL;DR

This paper proves the continuity of the drift of random walks on groups acting on metric spaces, including the mapping class group acting on Teichmüller space, and also shows the continuity of asymptotic entropy.

Contribution

It establishes the continuity of the drift with respect to transition measures under certain conditions and extends results to the mapping class group and entropy.

Findings

Drift varies continuously with transition probabilities.

Continuity of drift for sequences converging to Dirac measures.

Asymptotic entropy also varies continuously.

Abstract

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 $\mathrm{MCG}(S)$ acting on the Teichm\"uller 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 $\mathrm{MCG}(S)$ varies continuously.