Induced subsystems associated to a Cantor minimal system

Volume 117 / 2009

Heidi Dahl, Mats Molberg Colloquium Mathematicum 117 (2009), 207-221 MSC: 54H20, 37B10, 19K14. DOI: 10.4064/cm117-2-4


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)].


  • Heidi DahlDepartment of Mathematical Sciences
    Norwegian University of Science and Technology
    N-7491 Trondheim, Norway
  • Mats MolbergEducation and Natural Sciences
    Hedemark University College
    N-2418 Elverum, Norway

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