Coboundaries and eigenvalues of finitary S-adic systems

Abstract

An S-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary S-adic dynamical system is one where the directive sequence consists of morphisms selected from a finite set. We study eigenvalues and coboundaries for finitary recognizable S-adic dynamical systems, i.e., those where points can be uniquely desubstituted using the given sequence of morphisms. To do this we identify the notions of straightness and essential words, and use them to define a coboundary, inspired by of Host's formalism, which allows us to express necessary and sufficient conditions that a complex number must satisfy in order to be a continuous or measurable eigenvalue. We then apply our results to finitary directive sequences of substitutions of constant length, and show how to create constant-length $S$-adic shifts with non-trivial coboundaries. We show that in this case all continuous eigenvalues are rational and we give a complete description of the rationals that can be an eigenvalue, indicating how this leads to a Cobham-style result for these systems.

Figures (2)

• Figure 1: In this example, we construct $\mathcal{T}_n(a)$ with $\sigma_{n-1}(a)= cbd$, by concatenating portions of the towers of level $n-1$ for $c$, $b$ and then $d$.
• Figure 2: Part of the natural Bratteli diagram for a strongly straight directive sequence, where the letters $a$ and $b$ determine the same right-infinite limit word. Dashed edges correspond to minimal edges from $a$ or $b$ which lead directly to the limit word ${\boldsymbol a}={\boldsymbol b}$. The solid edges depict the limit word ${\boldsymbol a}$.

Theorems & Definitions (91)

• Proposition 2.1
• proof
• Definition 2.2: Transition and return words
• Definition 2.3
• Definition 2.4: Dynamical recognizability, $\sigma$-representations and recognizable directive sequences
• Example 2.5
• Theorem 2.6
• Example 2.7
• Theorem 2.8
• Lemma 2.9