Tameness, nullness, and amorphic complexity of 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.