Aperiodic substitutional systems and their Bratteli diagrams

TL;DR

This paper characterizes when aperiodic substitutional systems are equivalent to Vershik maps on stationary Bratteli diagrams, establishing conditions for homeomorphism, recognizability, and properties like expansiveness.

Contribution

It provides a complete characterization of aperiodic substitutional systems in terms of Bratteli diagrams and Vershik maps, including recognizability and conditions for homeomorphism.

Findings

Vershik homeomorphism is homeomorphic to an aperiodic substitutional system iff no minimal component is an odometer.

Every substitutional system with nesting property is homeomorphic to a Vershik map on a stationary Bratteli diagram.

All aperiodic substitutional systems are recognizable.

Abstract

In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $\phi$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional system if and only if no restriction of $\phi$ 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.