论文标题
熵，超纤维和泊松边界
Entropy, Ultralimits and the Poisson boundary
论文作者
论文摘要
在本文中，我们介绍了一组$ g $的超级措施阶级保存行动的概念，并表明它的furstenberg-Poisson边界可以作为自身的超级行动（配备适当选择的措施）获得。我们使用此结果来开始进行系统的定量研究，以了解基本问题与不变的距离有多近，可以找到$ g $ - 空间的措施，尤其是对于该组本身的行动。作为应用程序，我们表明，在可及的组上，关于信息理论的kullback-leibler差异（以及更一般而言，任何$ f $ divergence）总是存在“几乎不变的措施”，这是利用具有琐碎边界的度量的存在。更有趣的是，对于自由组$ f $和对称措施$λ$支持的发电机，可以明确计算furstenberg熵$h_λ（f，η）$的所有度量的$η$。令人惊讶的是，在发电机上的统一度量的情况下，该值与同一度量$λ$的furstenberg-Poisson边界的furstenberg熵相同，通常它是furstenberg furstenberg-poisson endropy furstenberg-poisson of to furstenberg-poisson to y v $ fulstenberg-poisson cub $λ$ niffent $ f $。
In this paper we introduce for a group $G$ the notion of ultralimit of measure class preserving actions of it, and show that its Furstenberg-Poisson boundaries can be obtained as an ultralimit of actions on itself, when equipped with appropriately chosen measures. We use this result in embarking on a systematic quantitative study of the basic question how close to invariant one can find measures on a $G$-space, particularly for the action of the group on itself. As applications we show that on amenable groups there are always "almost invariant measures" with respect to the information theoretic Kullback-Leibler divergence (and more generally, any $f$-divergence), making use of the existence of measures with trivial boundary. More interestingly, for a free group $F$ and a symmetric measure $λ$ supported on its generators, one can compute explicitly the infimum over all measures $η$ on $F$ of the Furstenberg entropy $h_λ(F,η)$. Somewhat surprisingly, while in the case of the uniform measure on the generators the value is the same as the Furstenberg entropy of the Furstenberg-Poisson boundary of the same measure $λ$, in general it is the Furstenberg entropy of the Furstenberg-Poisson boundary of a measure on $F$ different from $λ$.
