Elad Sayag | Searcharxiv

arXiv2024

Ultralimits, Amenable actions and Entropy

In this paper we show that the minimal value of Furstenberg entropy (along all measures, not restricting to stationary ones) for any amenable action is the same as for the action of the group on itself. Using the boundary amenability result of Adams, this allows us to compute the minimal value of the entropy over all the measure classes in the boundary of the free group. Similar results are proved for the action of a hyperbolic group on its Gromov boundary. Our main tool is an ultralimit realization of the Poisson boundary of a time dependent matrix-valued random walk on the group. This extends and refines the results and tools of previous paper of the author with Y. Shalom.

math.GR↗

arXiv2023

Entropy, Ultralimits and the Poisson boundary

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 $λ$.

math.GR↗