论文标题
Furstenberg系统的独特性
Uniqueness of a Furstenberg system
论文作者
论文摘要
给定一个可数的Amenable组$ g $，afølner序列$（f_n）\ subseteq g $，以及$ \ bar {d} _ {（f_n）}（e）= \ limsup_ = \ limsup_ { f_n |} {| f_n |}> 0 $，furstenberg的通信原理与一对$（e，（f_n））$一个度量保存系统$（x，x，\ nathcal {b}，μ，μ，（t_g） $μ（a）= \ bar {d} _ {（f_n）}（e）$，以一种方式，以使所有$ r \ in \ mathbb {n} $以及所有$ g_1，\ dots，g_r \ in g $ in G $ in G $ in G $ in G $ in G $ in G $ g_r^{ - 1} e）\geqμ（（t_ {g_1}）^{ - 1} a \ cap \ dots \ cap（t_ {g_r}）^{ - 1} a）$。我们表明，在某些自然假设下，系统$（x，\ mathcal {b}，μ，（t_g）_ {g \ in g}）$是唯一的，却是可测量的同构。我们还建立了这种唯一性结果的变体，用于非计算离散正式的半群以及广义的对应原理，该原理涉及有限的有限函数家族$ f_1，\ dots，f _ {\ ell}：g \ rightarrow \ rightArrow \ rightarrow \ mathbb {c} $。
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 $(X,\mathcal{B},μ,(T_g)_{g \in G})$ and a set $A \in \mathcal{B}$ with $μ(A)=\bar{d}_{(F_N)}(E)$, in such a way that for all $r \in \mathbb{N}$ and all $g_1,\dots,g_r \in G$ one has $\bar{d}_{(F_N)}(g_1^{-1}E \cap \dots \cap g_r^{-1}E)\geqμ((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show that under some natural assumptions, the system $(X,\mathcal{B},μ,(T_g)_{g \in G})$ 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 $f_1,\dots,f_{\ell}: G \rightarrow \mathbb{C}$.
