Let (X,T) be a Cantor minimal system and let (R,) 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,T̃), where T̃ 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 -actions on the Cantor set by T. Giordans et al. [J. Amer. Math. Soc. 21 (2008)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-4,
author = {Heidi Dahl and Mats Molberg},
title = {Induced subsystems associated to a Cantor minimal system},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {207-221},
zbl = {1182.54044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-4}
}
Heidi Dahl; Mats Molberg. Induced subsystems associated to a Cantor minimal system. Colloquium Mathematicae, Tome 116 (2009) pp. 207-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-4/