On continuity of drifts of the mapping class group

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