Decidable problems in substitution shifts

TL;DR

The paper extends substitution shift theory to the most general, possibly erasing, morphisms and proves the decidability of core dynamical properties such as aperiodicity, recognizability, irreducibility (under extra hypotheses), and minimality. It develops a comprehensive machinery around endomorphisms, higher block presentations, and derivation trees to analyze fixed points, periodic points, and growth behavior, establishing decidability for languages $ ext{L}( au)$ and $ ext{L}( ext{X}( au))$ and describing the finite structure of fixed and quasi-fixed points. A key contribution is showing that there is a finite computable set of non-growing-letter orbits and that every substitution shift is quasi-minimal with a finite subshift structure, along with a generalization that minimal shifts are conjugate to primitive ones. Collectively, these results yield algorithmic tools for understanding a broad class of substitution shifts and have implications for automatic sequences and numeration systems, while highlighting open questions about decidability of conjugacy and logical formalisms for such systems.

Abstract

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the shift space generated by iteration, such as aperiodicity, recognizability and (under an additional assumption) irreducibility, or minimality.

Figures (10)

• Figure 2.1: The $k$-th higher block code.
• Figure 2.2: The 2nd-higher block presentation of $\{a,b\}^\mathbb{Z}$.
• Figure 3.1: The tree $T_\sigma(a,2)$.
• Figure 3.2: The graph $G(\sigma)$.
• Figure 3.3: Decidability of $\mathcal{L}(\mathsf X(\sigma))$

Theorems & Definitions (41)

• Example 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Example 2.5
• Example 3.1
• Example 3.2
• Proposition 3.3
• Example 3.4
• Example 3.5