Aperiodic substitutional systems and their Bratteli diagrams

Abstract

In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $\varphi$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional system if and only if no restriction of $\varphi$ to a minimal component is homeomorphic to an odometer. We also show that every aperiodic substitutional system generated by a substitution with nesting property is homeomorphic to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitutional system is recognizable. The classes of $m$-primitive substitutions and associated to them derivative substitutions are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

42 pages