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On continuity of drifts of the mapping class group
Mathematical Research Letters ( IF 1 ) Pub Date : 2021-05-01 , DOI: 10.4310/mrl.2021.v28.n3.a8
Hidetoshi Masai 1
Affiliation  

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.

中文翻译:

关于映射类组漂移的连续性

作用于度量空间 $X$ 的可数组 $G$ 上的随机游走给出了称为漂移的特性,该特性仅取决于随机游走的转移概率度量 $\mu$。漂移是随机游走的“平移距离”。在本文中,我们证明了漂移随转移概率度量连续变化,假设 $X$ 上的距离和水平函数由某些比率表示。例如,我们考虑作用于 Teichmüller 空间的映射类组 MCG($S$)。通过使用南北动力学,我们还考虑了收敛到狄拉克测度的序列的漂移的连续性。作为附录,我们证明了 MCG($S$) 上随机游走的渐近熵是连续变化的。
更新日期:2021-06-02
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