Uniqueness of a Furstenberg system

Abstract

Given a countable amenable group $G$, a Følner sequence $(F_N)\subseteq G$, and a set $E\subseteq G$ with ${\bar{d}}_{(F_N)}(E)={\mathrm{limsup}}_{N\to \infty }\frac{|E\cap F_N|}{|F_N|}>0$, Furstenberg's correspondence principle associates with the pair $(E,(F_N))$ a measure preserving system $(X,\mathcal{B},\mu ,(T_g{)}_{g\in G})$ and a set $A\in \mathcal{B}$ with $\mu (A)={\bar{d}}_{(F_N)}(E)$, in such a way that for all $r\in \mathbb{N}$ and all $g_1,\dots ,g_r\in G$ one has ${\bar{d}}_{(F_N)}(g_1^{-1}E\cap \cdots \cap g_r^{-1}E)\ge \mu ((T_{g_1}{)}^{-1}A\cap \cdots \cap (T_{g_r}{)}^{-1}A)$. We show that under some natural assumptions, the system $(X,\mathcal{B},\mu ,(T_g{)}_{g\in G})$ is unique up to a measurable isomorphism. We also establish variants of this uniqueness result for non-countable discrete amenable semigroups as well as for a generalized correspondence principle which deals with a finite family of bounded functions $f_1,\dots ,f_{\ell }:G\to \mathbb{C}$.

14 pages