Tameness, nullness, and amorphic complexity of automatic systems

TL;DR

This paper characterizes tameness and nullness in minimal automatic systems generated by primitive substitutions using amorphic complexity, providing a computable invariant that distinguishes these properties.

Contribution

It offers a complete characterization of tameness and nullness in automatic systems via amorphic complexity, including a closed-form formula for this invariant.

Findings

Tameness and nullness correspond to amorphic complexity value of one.

A closed-form formula for amorphic complexity in primitive substitution systems.

Tameness and nullness are equivalent in infinite automatic systems.

Abstract

In topological dynamics, tame and null systems arise naturally in the study of low-complexity aperiodic behaviour, yet providing concrete and easily testable conditions to establish their existence in a canonical class of systems is often nontrivial. We give a complete characterisation of tameness and nullness for minimal automatic systems generated by primitive constant length substitutions in terms of amorphic complexity -- a numerical invariant recently introduced to study zero entropy systems. We derive an easily computable closed formula for this invariant in this setting and show that, for infinite automatic systems, tameness and nullness are equivalent to its value being one.