Induced subsystems associated to a Cantor minimal system
Volume 117 / 2009
Colloquium Mathematicum 117 (2009), 207-221
MSC: 54H20, 37B10, 19K14.
DOI: 10.4064/cm117-2-4
Abstract
Let $(X,T)$ be a Cantor minimal system and let $(R, \mathcal{T})$ be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system $(Y,S)$ there exists a closed subset $Z$ of $X$ such that $(Y,S)$ is conjugate to the subsystem $(Z,\widetilde{T})$, where $\widetilde{T}$ is the induced map on $Z$ from $T$. We explore when we may choose $Z$ to be a $T$-regular and/or a $T$-thin set, and we relate $T$-regularity of a set to $R$-étaleness. The latter concept plays an important role in the study of the orbit structure of minimal $\mathbb{Z}^d$-actions on the Cantor set by T. Giordans et al. [J. Amer. Math. Soc. 21 (2008)].
