Furstenberg discretization with moment control and entropy rigidity
Abstract
Let G be a locally compact group, let Γ be a lattice in G, and let η be a spread-out probability measure on G. A Furstenberg discretization of η to Γ is a probability measure μ on Γ such that every (G,η)-stationary space is also (Γ,μ)-stationary. For cocompact lattices Γ and under mild regularity conditions on η, we construct Furstenberg discretizations satisfying finite moment conditions of arbitrary order with respect to arbitrary prescribed subadditive functions. Our approach relies on a rather general study of differential entropy and Lyapunov exponents in the setting of locally compact groups. We relate Lyapunov exponents to spectral radii in weighted L1-algebras and prove vanishing results for weights coming from length functions. Additionally, we prove that if G has the property of rapid decay or the Kunze--Stein property, this yields entropy formulas for stationary actions whose Koopman representation is weakly contained in the left-regular representation. As consequences, we obtain preservation results for Poisson boundaries under discretization and characterize Zimmer amenability of stationary actions in terms of weak containment and maximal Furstenberg entropy.
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BibTeX
@misc{andersonsackaney2026furstenberg,
title = {{Furstenberg discretization with moment control and entropy rigidity}},
author = {Benjamin Anderson-Sackaney and Tim de Laat and Ebrahim Samei and Matthew Wiersma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12268},
eprint = {2610.12268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12268},
}
RIS
TY - UNPB
TI - Furstenberg discretization with moment control and entropy rigidity
AU - Benjamin Anderson-Sackaney
AU - Tim de Laat
AU - Ebrahim Samei
AU - Matthew Wiersma
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12268
UR - https://arxiv.org/abs/2610.12268
AB - Let G be a locally compact group, let Γ be a lattice in G, and let η be a spread-out probability measure on G. A Furstenberg discretization of η to Γ is a probability measure μ on Γ such that every (G,η)-stationary space is also (Γ,μ)-stationary. For cocompact lattices Γ and under mild regularity conditions on η, we construct Furstenberg discretizations satisfying finite moment conditions of arbitrary order with respect to arbitrary prescribed subadditive functions. Our approach relies on a rather general study of differential entropy and Lyapunov exponents in the setting of locally compact groups. We relate Lyapunov exponents to spectral radii in weighted L^1-algebras and prove vanishing results for weights coming from length functions. Additionally, we prove that if G has the property of rapid decay or the Kunze--Stein property, this yields entropy formulas for stationary actions whose Koopman representation is weakly contained in the left-regular representation. As consequences, we obtain preservation results for Poisson boundaries under discretization and characterize Zimmer amenability of stationary actions in terms of weak containment and maximal Furstenberg entropy.
ER -