Induced subsystems associated to a Cantor minimal system

Heidi Dahl¹ ; Mats Molberg²

¹Department of Mathematical Sciences Norwegian University of Science and Technology N-7491 Trondheim, Norway
²Education and Natural Sciences Hedemark University College N-2418 Elverum, Norway

Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 207-221

Résumé

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)].

Détail

DOI : 10.4064/cm117-2-4

Keywords: cantor minimal system mathcal associated tale equivalence relation orbit equivalence relation arbitrary cantor minimal system there exists closed subset conjugate subsystem widetilde where widetilde induced map explore may choose t regular t thin set relate t regularity set r taleness latter concept plays important role study orbit structure minimal mathbb d actions cantor set giordans amer math soc