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 $\bar {d}_{(F_N)}(E)=\limsup _{N \to \infty } \frac {|E \cap F_N|}{|F_N|}>0$, Furstenberg’s correspondence principle associates with the pair $(E,(F_N))$ a measure preserving system $\mathbb {X}=(X,\mathcal {B},\mu ,(T_g)_{g \in G})$ 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 $\bar {d}_{(F_N)}(g_1^{-1}E \cap \dots \cap g_r^{-1}E)\geq \mu ((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show that under some natural assumptions, the system $\mathbb {X}$ 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 $f_1,\dots ,f_{\ell }: G \rightarrow \mathbb {C}$.