2022/02/14 by Elad Sayag, Yehuda Shalom, Sayag, Elad +1
Computer Science · Mathematics · #20F65 #22F10 #28D20 #31C05 #46B08 #46M07 #60G10 #60J50 #82C41 #94A17 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2202.06607
openalex publication_date 2022/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 hλ(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.