标题： 汤普森群 $T$ 的泊松边界不是圆圈 显示英文标题

标题： The Poisson boundary of Thompson's group $T$ is not the circle

摘要： 设$\mu$是定义在圆周上保向同胚的可数群$G \leq \mathrm{Homeo}_+(S^1)$上的一个非退化概率测度，并且该测度具有有限熵，在$S^1$上作用时是近似的、极小的并且拓扑上非自由的。我们证明了圆周$S^1$赋予其唯一的$\mu$-平稳概率测度后，不是$(G,\mu)$的泊松边界。 当$G$是汤普森群$T$且$\mu$是有限支撑时，这就回答了 B. Deroin [遍历理论动力系统, 2013] 和 A. Navas [国际数学家大会论文集, 2018] 提出的问题。 显示英文摘要

摘要： Let $\mu$ be a nondegenerate probability measure with finite entropy on a countable group $G \leq \mathrm{Homeo}_+(S^1)$ of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on $S^1$. We prove that the circle $S^1$ endowed with its unique $\mu$-stationary probability measure is not the Poisson boundary of $(G,\mu)$. When $G$ is Thompson's group $T$ and $\mu$ is finitely supported, this answers a question posed by B. Deroin [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].