AI 中文总结
本文研究仿射评估映射（AEM），构造了与良态AEM对应的单维数群，利用相关定理证明其可由极小康托尔同胚实现，并将该框架推广到可数可和群的极小康托尔作用。
AI 中文摘要
设$X$为康托尔空间，$Q\subset M_{fc}(X)$是无原子全支撑概率测度的紧Choquet单形。我们引入相关的**仿射评估映射（Affine Evaluation Map, AEM）**，它为每个 clopen 集$A\subset X$分配仿射函数$\widehat A(\mu)=\mu(A)$（其中$\mu\in Q$），并研究通过在$Q$上逐点比较这些评估函数得到的几何子集条件。对于一个良态几何AEM，我们构造有序群$G_Q=C(X,\mathbb Z)/N_Q$，其中$N_Q=\left\{f\in C(X,\mathbb Z): \int f\\,d\mu=0\\ \text{对所有}\\ \mu\in Q\right\}$，证明它是一个单维数群，其归一化状态空间恰好是$Q$。我们证明其序区间$[0,u]$正是 clopen 尺度，且$J_Q=N_Q$，其中$J_Q$由满足$\widehat A=\widehat B$的基本关系$\mathbf 1_A-\mathbf 1_B$生成。我们还证明全稳定子$\mathcal H_Q$的不变测度单形恰好是$Q$。利用 clopen 尺度性质与Herman–Putnam–Skau实现定理，我们得到一个维数群证明：每个良态几何AEM都可由极小康托尔同胚$T$实现，满足$Q=M_T(X)$。对于康托尔极小系统，我们将$G_Q$与经典维数群模无穷小量等同起来。我们还通过具体例子区分测度稳定子的良态性、公平性、遍历性与极小性。最后，我们将AEM框架应用于可数可和群的极小康托尔作用。若$Q=M_G(X)$，则$Q$自然定义一个恰当几何AEM，我们阐明前述理论中哪些部分仅依赖于$Q$，哪些是针对$\mathbb Z$-动力系统特有的。特别地，$Q$是良态的当且仅当存在$X$的极小同胚$T$使得$M_T(X)=M_G(X)$。
英文摘要
Let $X$ be a Cantor space and let $Q\subset M_{fc}(X)$ be a compact Choquet simplex of atomless full-support probability measures. We introduce the associated \emph{Affine Evaluation Map} (AEM), which assigns to each clopen set $A\subset X$ the affine function \[ \widehat A(μ)=μ(A),\qquad μ\in Q, \] and study the geometric subset condition obtained by comparing these evaluation functions pointwise on $Q$. For a good geometric AEM we construct the ordered group \[ G_Q=C(X,\mathbb Z)/N_Q, \qquad N_Q=\left\{f\in C(X,\mathbb Z): \int f\,dμ=0\ \text{for every }μ\in Q\right\}, \] and show that it is a simple dimension group whose normalized state space is canonically $Q$. We prove that its order interval $[0,u]$ is precisely the clopen scale and that \[ J_Q=N_Q, \] where $J_Q$ is generated by the elementary relations $\mathbf 1_A-\mathbf 1_B$ with $\widehat A=\widehat B$. We also show that the full stabilizer $\mathcal H_Q$ has invariant-measure simplex exactly $Q$. Using the clopen-scale property and the Herman--Putnam--Skau realization theorem, we obtain a dimension-group proof that every good geometric AEM is realized by a minimal Cantor homeomorphism $T$ with $Q=M_T(X)$. For a Cantor minimal system we identify $G_Q$ with the classical dimension group modulo infinitesimals. We further distinguish goodness, fairness, ergodicity, and minimality of the measure stabilizer by explicit examples. Finally, we apply the AEM framework to minimal Cantor actions of countable amenable groups. If $Q=M_G(X)$, then $Q$ canonically defines a proper geometric AEM, and we clarify which parts of the preceding theory depend only on $Q$ and which are specifically $\mathbb Z$-dynamical. In particular, $Q$ is good if and only if there exists a minimal homeomorphism $T$ of $X$ such that \[ M_T(X)=M_G(X). \]
