熵、超极限与泊松边界
Entropy, Ultralimits and the Poisson boundary
AI总结：
本文引入群作用的超极限概念，证明Furstenberg-Poisson边界可由超极限获得，并用于定量研究近不变测度，给出可解群上几乎不变测度的存在性及自由群上Furstenberg熵下确界的显式公式。
摘要：
在本文中，我们为群$G$引入了其保测类作用的超极限概念，并证明当配备适当选择的测度时，其Furstenberg-Poisson边界可以通过对$G$自身作用的超极限获得。我们利用这一结果开始了对基本问题的系统性定量研究：在$G$-空间上，可以找到多接近不变的测度，特别是对于群对自身的作用。作为应用，我们证明在可解群上，关于信息论Kullback-Leibler散度（更一般地，任何$f$-散度）总是存在“几乎不变测度”，这利用了具有平凡边界的测度的存在性。更有趣的是，对于自由群$F$和支撑在其生成元上的对称测度$λ$，可以显式计算所有$F$上测度$η$的Furstenberg熵$h_λ(F,η)$的下确界。有点令人惊讶的是，虽然在生成元上的均匀测度的情况下，该值与同一测度$λ$的Furstenberg-Poisson边界的Furstenberg熵相同，但一般来说，它是$F$上不同于$λ$的测度的Furstenberg-Poisson边界的Furstenberg熵。
英文摘要：
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 $λ$.
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