Directive words of episturmian words: equivalences and normalization

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.

💡 Research Summary

The paper investigates the relationship between episturmian words—a broad generalization of Sturmian sequences to arbitrary finite alphabets—and the infinite “directive words” that encode their construction via pure episturmian morphisms. The authors first recall the necessary background: an episturmian word is an infinite word whose set of factors is closed under reversal and which has at most one right (or left) special factor of each length. Such words can be generated by repeatedly applying morphisms Lₐ (which prepend a to every letter different from a) and Rₐ (which append a to every letter different from a). A “directive word” is an infinite sequence of letters equipped with a spin (L or R) that tells which morphism to apply at each step; the infinite product of these morphisms yields the episturmian word.

The central problem addressed is: when do two (finite or infinite) spinned words direct the same episturmian morphism or the same episturmian word? Earlier work introduced a “block‑equivalence” notion for finite directive words, but this approach does not fully capture the situation for infinite or aperiodic words. The authors therefore develop a more robust equivalence concept and, crucially, a normalization procedure for directive words.

The normalization theorem (Theorem 5.2) shows that any directive word can be transformed, by a series of elementary spin‑rearrangement and compression rules, into a unique “normalized” form. The rules are:

1. Move any block of consecutive R‑spins as far left as possible without crossing preceding L‑spins.
2. Collapse consecutive L‑spins on the same letter into a single L‑spin.
3. Ensure that for each letter a, the first occurrence in the directive word carries an L‑spin, while all later occurrences carry R‑spins.

The resulting normalized directive word is canonical: any two directive words that generate the same episturmian word will have identical normalizations. This eliminates the need for block‑equivalence and provides a direct, algorithmic way to decide whether two directive words are equivalent.

Using this canonical form, the authors characterize episturmian words with a unique directive word. They prove that uniqueness occurs precisely in two cases:

• The normalized directive word consists entirely of L‑spins (so each letter appears first with an L‑spin and never again with an R‑spin), or
• It consists entirely of R‑spins and the set of letters that appear infinitely often in the underlying (unspinned) word has cardinality one.

These conditions generalize the known fact that Sturmian words (the binary case) have a unique directive word. Moreover, the paper distinguishes periodic from aperiodic episturmian words via the structure of the normalized directive word: periodic words correspond to directives of the form μ_{ŵ}(x)ω (a finite spinned word followed by infinite repetitions of a single letter), whereas aperiodic words exhibit an infinite, non‑repeating pattern of L‑ and R‑spins.

An important application is the characterization of quasiperiodic episturmian words. By examining the normalized directive words, the authors show that a word is quasiperiodic if and only if its directive word can be expressed as a finite spinned prefix followed by an infinite repetition of a spinned block, extending earlier results for Sturmian and Arnoux‑Rauzy sequences.

The paper concludes with suggestions for future work, including extending the normalization framework to infinite alphabets, exploring connections with S‑adic dynamical systems, and using normalized directives to classify episturmian morphisms up to isomorphism.

Overall, the work provides a comprehensive and constructive answer to the equivalence problem for directive words, introduces a practical normalization algorithm, and leverages it to obtain new structural insights into episturmian words, their periodicity, and quasiperiodicity.