Quasi-fixed points of substitutive systems

TL;DR

This paper classifies automatic sequences within automatic systems generated by constant length substitutions, revealing their structure through quasi-fixed points and extending to factor maps, with conjectures for broader cases.

Contribution

It provides a complete classification of automatic sequences in substitution systems using quasi-fixed points, extending understanding of their structure and arithmetic properties.

Findings

Most sequences in an automatic system are not automatic.

Automatic sequences in a system are characterized by quasi-fixed points.

The classification extends to factor maps between systems.

Abstract

We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within an automatic system are not themselves automatic. We provide a complete and succinct classification of automatic sequences that lie in a given automatic system in terms of the quasi-fixed points of the substitution defining the system. Our result extends to factor maps between automatic systems and highlights arithmetic properties underpinning these systems. We conjecture that a similar statement holds for general nonconstant length substitutions.