发表机构
University of Victoria; Kiel University; University of Saskatchewan; University of Winnipeg（维多利亚大学; 基尔大学; 萨斯喀彻温大学; 温尼伯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对局部紧群的余紧格，本文构造了满足任意阶矩条件的Furstenberg离散化，关联了Lyapunov指数与谱半径，推导了相关熵公式，得到Poisson边界保持结果并刻画了平稳作用的Zimmer可和性。
AI 中文摘要
设$G$为局部紧群，$\boldsymbol{\text{Γ}}$为$G$中的格，$\boldsymbol{\text{η}}$为$G$上的扩散概率测度。将$\boldsymbol{\text{η}}$离散化为$\boldsymbol{\text{Γ}}$上的Furstenberg离散化，指的是$\boldsymbol{\text{Γ}}$上的概率测度$\boldsymbol{\text{μ}}$，使得每个$(G,\boldsymbol{\text{η}})$平稳空间也为$(\boldsymbol{\text{Γ}},\boldsymbol{\text{μ}})$平稳空间。对于余紧格$\boldsymbol{\text{Γ}}$及$\boldsymbol{\text{η}}$满足的温和正则条件，我们构造了满足关于任意指定次可加函数的任意阶有限矩条件的Furstenberg离散化。我们的方法依赖于对局部紧群框架下的微分熵与Lyapunov指数的一般性研究，将Lyapunov指数与加权$L^1$代数中的谱半径关联，并证明了来自长度函数的权重的消失结果。此外，我们证明若$G$具有快速衰减性质或Kunze–Stein性质，这会为其Koopman表示弱包含于左正则表示的平稳作用生成熵公式。作为推论，我们得到离散化下Poisson边界的保持结果，并通过弱包含与极大Furstenberg熵刻画平稳作用的Zimmer可和性。
英文摘要
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.
