Universal minimal topological dynamical systems (Q2480569)

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy \(< \log \alpha \), admit a generating partition with no more than \(\alpha\) sets. In symbolic dynamics terminology, these results can be phrased as follows: \({\mathbb N}^{\mathbb Z}\) is a universal system in the category of aperiodic systems, and \([\alpha ]^{\mathbb Z}\) is a universal system for aperiodic ergodic systems with entropy \(< \log \alpha \). Weiss (1989) presented a Minimal system, on a Compact space (a subshift of \({\mathbb N}^{\mathbb Z}\)) which is universal for aperiodic systems. In the paper a joint generalization of both results is presented. Main results of the paper: 1. There exists a minimal subshift \(M\) of \(\{ 0,1\}^{\mathbb Z} \) which is universal for all standard aperiodic ergodic dynamical systems with entropy \(0\). 2. Fix some \(\varepsilon >0\). There exists a minimal subshift \(M \subset \{1,\dots, \alpha\}^{\mathbb Z} \) which is universal for aperiodic standard ergodic systems with entropy \(< (\log \alpha - \varepsilon)\).