Affine Evaluation Maps, Dimension Groups, and Fair measures

Abstract

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 $1_A-1_B$ with $\hat{A}=\hat{B}$. 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 \[ M_T(X)=M_G(X). \]