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Tameness, nullness, and amorphic complexity of automatic systems

Maik Gröger, Elżbieta Krawczyk

TL;DR

This work gives 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.

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.

Tameness, nullness, and amorphic complexity of automatic systems

TL;DR

This work gives 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.

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.
Paper Structure (16 sections, 22 theorems, 111 equations)

This paper contains 16 sections, 22 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. Asymptotic notations
  4. Topological and symbolic dynamics
  5. Tameness and nullness
  6. Pseudometrics
  7. Box dimension
  8. Amorphic complexity
  9. Substitutions
  10. Systems defined by substitutions
  11. Induced pseudometrics and discrepancy substitutions
  12. Pseudometrics related to clopen partitions
  13. Discrepancy substitution
  14. Pseudometrics related to pairs of letters of maximal growth
  15. Amorphic complexity of constant length substitution shifts
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose $\varphi\colon {\mathcal{A}}\to{\mathcal{A}}^*$ is a primitive substitution of length $k\geqslant 2$. Then, the following are equivalent:

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7: FuhrmannGroeger2020
  • Remark 2.8
  • ...and 50 more