Entropy, ultralimits and the Poisson boundary

Abstract

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 of 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 $\lambda$ supported on its generators, one can compute explicitly the infimum over all measures $\eta$ on $F$ of the Furstenberg entropy $h_{\lambda }(F,\eta )$. 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 $\lambda$, in general it is the Furstenberg entropy of the Furstenberg–Poisson boundary of a measure on $F$ different from $\lambda$.