Automaticity of uniformly recurrent substitutive sequences

TL;DR

This paper characterizes when uniformly recurrent substitutive sequences are automatic using incidence matrices, resolving a recent open question and extending the criterion to minimal substitutive systems.

Contribution

It provides a complete criterion for automaticity of uniformly recurrent substitutive sequences based on incidence matrices, and constructs a minimal system with unique factor properties.

Findings

Criterion for automaticity based on incidence matrices

Resolution of a question by Allouche, Dekking, and Queffélec

Construction of a minimal system with uncountable-to-one factor map

Abstract

We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queff\'elec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems. Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer.