Automaticity of uniformly recurrent substitutive sequences

TL;DR

The paper delivers a complete dynamical characterization of automaticity for uniformly recurrent substitutive sequences by linking it to the return substitution of the underlying primitive sequence: a fixed point $x$ of a primitive substitution is automatic if and only if the left vector $^{t}(|\varphi^{s}(a)|)_{a\in \mathrm{R}_a}$ is an eigenvector of the return-substitution incidence matrix $M_{\tau}$, where $s$ is the size of the largest Jordan block for eigenvalue $0$. The proof combines a Dekking-type sufficiency criterion, recognizability to pass to a constant-length substitution, and a reduction to two-sided and one-sided cases, with a key role for the conjugacy result of MY-21. The authors illustrate the boundary between automatic and non-automatic behavior through examples and construct a minimal substitutive system with ${\mathbf Z}_2$ as its maximal equicontinuous factor whose fibre map is everywhere uncountable-to-one, motivating a broader conjecture that automaticity corresponds to finite-to-one extensions of $k$-adic odometers. Collectively, the work bridges combinatorial substitution theory with dynamical systems, yielding algorithmic criteria and shedding light on fibre structures over maximal equicontinuous factors, with implications for the dynamic characterisation of automaticity.

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 Queffélec 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.

Figures (5)

• Figure 1: The first four levels of the tree in $T_{\varphi}$ rooted at $\varphi[a]$, showing 8 out of 16 nodes in level 2 and 8 out of 64 nodes in level 3.
• Figure 2: The complete first 4 levels of the subtree of $T_{\varphi}$ starting with $\varphi[a]$ corresponding to the label $l=0^{\infty}$.
• Figure 3: The children of $T^i\varphi^n[a]$ for $0\leqslant i<\left| \varphi^n(a) \right|$ with their labels.
• Figure 4: The children of $T^i\varphi^n[b]$ for $0\leqslant i<\left| \varphi^n(b) \right|$ with their labels.
• Figure 5: The complete first 4 levels of the tree with labels 0 starting with $\tau[a]$ for the Thue--Morse substitution $\tau$. Here the nodes $T^4\tau^2[b]$, $T^{12}\tau^2[a]$, $T^8\tau^3[b]$, $T^{24}\tau^3[a]$, $T^{40}\tau^3[a]$, $T^{56}\tau^3[b]$ are leaves, i.e. they have no continuation with label 0.

Theorems & Definitions (33)

• Theorem 2
• Example 3
• Theorem 4
• Conjecture 5
• Corollary 1.1
• Example 1.2
• Corollary 1.3
• proof
• Example 1.4
• Example 1.5