INDUCED SUBSYSTEMS ASSOCIATED TO A CANTOR MINIMAL SYSTEM

Let (X, T) be a Cantor minimal system and let (R,T) be the associated etale 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, (T) over tilde), where (T) over tilde 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-etaleness. The latter concept plays an important role in the study of the orbit structure of minimal Z(d)-actions on the Cantor set by T. Giordans et al. [J. Amer. Math. Soc. 21 (2008)].