On the minimal components of substitution subshifts

TL;DR

The paper gives a complete, computable description of the minimal components of substitution subshifts $X_\sigma$ on a finite alphabet. It separates the dynamics into tame and wild components: tame minimal components arise from main sub-substitutions on a suitable subalphabet of the growing letters, while wild minimal components are exactly the single-periodic orbits over bounded letters generated by left/right isolated letters, captured by computable words $LP(c)$ and $RP(c)$. A key technical achievement is showing that $\tilde{\sigma}$ induces a permutation on the set of minimal components, with this action encoded by explicit directed graphs; this yields effective counting and bounding results: $MC(\sigma)$ is computable and obeys tight bounds dependent on $|A|$ and the partition into growing and bounded letters. The authors provide an explicit algorithm and Python code to compute $B$, $C$, the minimal components, and their number, and illustrate the theory with all substitutions on two letters. The framework unifies growing and non-growing cases and offers a pathway to generalizations to irreducible components and broader subshift classes.

Abstract

In this paper we study substitutions on $A^\mathbb{Z}$ where $A$ is a finite alphabet. We precisely characterize the minimal components of substitution subshifts, give an optimal bound for their number and describe their dynamics. The explicitness of these results provides a method to algorithmically compute and count the minimal components of a given substitution subshift.

Figures (2)

• Figure 1: $a u b \sqsubset \sigma(a') \sqsubset \sigma^{k+1}(c)$.
• Figure 2: $aub \sqsubset \sigma(a'u'b') \sqsubset \sigma^{k+1}(c)$.

Theorems & Definitions (142)

• Example 1.1
• Remark 1.1
• Proposition 1.1: BPR2
• Definition 1.1
• Remark 1.2
• Example 1.2
• Definition 1.2
• Example 1.3
• Theorem 1.1: Shimomura
• Corollary 1.1: Shimomura