Embeddings of Relative Dimension Groups for Zero-Dimensional Dynamical Systems

Abstract

Let $\pi :(X,T)\to (Y,S)$ be a factor map between zero-dimensional dynamical systems, and let $X_0$ and $Y_0$ be quasi-sections with $\pi (X_0)=Y_0$. We study the homomorphism ${\pi }^{\ast }:K^0(Y|Y_0,S)\to K^0(X|X_0,T)$, $[f]\mapsto [f\circ \pi ]$, between the corresponding relative dimension groups, and identify it with the homomorphism induced by the premorphism between Bratteli--Vershik realizations associated with $\pi$. Generalizing a result of Glasner and Weiss for minimal Cantor systems, we show that if $Y_0$ is quasi-simple, then ${\pi }^{\ast }$ is an injective unital order-preserving homomorphism, and that if $X_0={\pi }^{-1}(Y_0)$, then ${\pi }^{\ast }$ is a unital order embedding; the latter condition is characterized in terms of the premorphism. Examples show that quasi-simplicity cannot be dropped, and that without the condition $X_0={\pi }^{-1}(Y_0)$ the map ${\pi }^{\ast }$ need not be an order embedding.