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Obstructions to return preservation for episturmian morphisms

Valérie Berthé, Herman Goulet-Ouellet

TL;DR

The paper addresses whether primitive episturmian morphisms preserve return sets of factors and proves the existence of infinitely many obstructions, extending prior Sturmian results. It develops a framework based on Rauzy graphs, Palindromic closure, and the conjugacy index $ind(\sigma)$ alongside the minimal letter $a_{ ext{min}}$ to construct explicit obstruction families, depending on whether $ind(\sigma)\ge |\sigma(a_{ ext{min}})|$ or $ind(\sigma)\le\frac{\Vert\sigma\Vert-|A|}{|A|-1}-|\sigma(a_{ ext{min}})|$. The analysis provides precise obstructions within the two regimes and describes how return sets transform under episturmian morphisms via detailed return-word and Rauzy-graph descriptions. This work clarifies the limits of preservation-based methods for understanding return-set dynamics and motivates extending these techniques to broader dendric families to map the full boundary of the property.

Abstract

This paper studies obstructions to preservation of return sets by episturmian morphisms. We show, by way of an explicit construction, that infinitely many obstructions exist. This generalizes and improves an earlier result about Sturmian morphisms.

Obstructions to return preservation for episturmian morphisms

TL;DR

The paper addresses whether primitive episturmian morphisms preserve return sets of factors and proves the existence of infinitely many obstructions, extending prior Sturmian results. It develops a framework based on Rauzy graphs, Palindromic closure, and the conjugacy index alongside the minimal letter to construct explicit obstruction families, depending on whether or . The analysis provides precise obstructions within the two regimes and describes how return sets transform under episturmian morphisms via detailed return-word and Rauzy-graph descriptions. This work clarifies the limits of preservation-based methods for understanding return-set dynamics and motivates extending these techniques to broader dendric families to map the full boundary of the property.

Abstract

This paper studies obstructions to preservation of return sets by episturmian morphisms. We show, by way of an explicit construction, that infinitely many obstructions exist. This generalizes and improves an earlier result about Sturmian morphisms.
Paper Structure (7 sections, 20 theorems, 60 equations, 6 figures, 1 table)

This paper contains 7 sections, 20 theorems, 60 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First properties of episturmian morphisms
  4. Conjugacy classes of episurmian morphisms
  5. Rauzy graphs and return words
  6. Proof of the main result
  7. Conclusion

Key Result

Theorem 1.1

Let $\sigma$ be a primitive episturmian morphism over the alphabet $A$. Then there is a letter $a_{\min}$ in $A$ such that $\sigma$ fails the return preservation property for all but finitely many words in:

Figures (6)

  • Figure 1: The standard ternary tree.
  • Figure 2: Values of $\operatorname{Pal}$ over a ternary alphabet.
  • Figure 3: Rauzy graph $\Gamma_4$ in the "tetrabonacci" shift ($d$-bonacci with $d=4$).
  • Figure 4: Possible shapes of Rauzy graphs in a strict episturmian shift on 4 letters.
  • Figure 5: Rauzy graph $\Gamma_8$ from \ref{['e:dl']} with pairs $(\operatorname{d},\ell)$.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Theorem 1.1
  • Example 3.1: $d$-bonacci
  • Theorem 3.2: Justin2002
  • Example 3.3
  • Proposition 3.4
  • proof
  • Theorem 4.1: Richomme2003, Theorem 5.1
  • Definition 4.2
  • Lemma 4.3
  • proof
  • ...and 33 more