Minimal Cantor Type Sets on Discrete Dynamical Systems

Abstract

Cantor sets on topological spaces can be obtained as minimal sets with respect to discrete dynamical systems (d.d.s.), \((X, f)\) where X is a compact metric space and f could be an homeomorphism or a non-invertible continuous map. In general, given a Cantor type set or simply a Cantor set in a compact metric space, it is difficult to obtain a d.d.s. having it as minimal. Examples of Cantor sets that are minimal or non-minimal can be obtained using the ternary Cantor set on the real interval and constructing appropriate continuous maps. In the case of the minimality of such Cantor sets, every point of them are uniformly recurrent points for f. Other examples of minimal Cantor sets like the shift space \(\Sigma^{2}\) or subsets of it are given for homeomorphisms. The whole Smale horseshoe is non-minimal for an adequate homeomorphism or one of its subsets is minimal for it.