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 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.
Elad Sayag; Yehuda Shalom. Entropy, ultralimits and the Poisson boundary. Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 525-565. doi: 10.4171/ggd/887
@article{10_4171_ggd_887,
author = {Elad Sayag and Yehuda Shalom},
title = {Entropy, ultralimits and the {Poisson} boundary},
journal = {Groups, geometry, and dynamics},
pages = {525--565},
year = {2025},
volume = {19},
number = {2},
doi = {10.4171/ggd/887},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/887/}
}
TY - JOUR
AU - Elad Sayag
AU - Yehuda Shalom
TI - Entropy, ultralimits and the Poisson boundary
JO - Groups, geometry, and dynamics
PY - 2025
SP - 525
EP - 565
VL - 19
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/887/
DO - 10.4171/ggd/887
ID - 10_4171_ggd_887
ER -
