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$\mathcal{S}$-adic characterization of minimal dendric shifts

France Gheeraert, Julien Leroy

TL;DR

The paper advances the theory of dendric shifts by providing a general S-adic characterization for minimal dendric shifts over arbitrary alphabets, built from two finite graphs that encode left and right extension data. It proves that minimal dendric shifts are exactly those with primitive $ rak{S}$-adic representations whose morphisms are dendric return morphisms and that the representation labels infinite paths in both $ ull\mathcal{G}^L( rak{S})$ and $\mathcal{G}^R(\frak{S})$, yielding concrete criteria for deducing dendricity and eventual dendricity. The framework yields decidability results for eventual dendricity in substitutive settings and provides explicit constructions and examples, including a four-letter case with one right special factor per length and a connection to interval exchanges via planar extension graphs. The work also extends known ternary results to general alphabets and situates the theory within the dynamics of derived shifts, return morphisms, and Rauzy-type induction, suggesting broad avenues for future regular-language descriptions of dendric return morphisms and applications to IET codings.

Abstract

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal{S}$-adic representation where the morphisms in $\mathcal{S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper we give an $\mathcal{S}$-adic characterization of this family by means of two finite graphs. As an application, we are able to decide whether a shift space generated by a uniformly recurrent morphic word is (eventually) dendric.

$\mathcal{S}$-adic characterization of minimal dendric shifts

TL;DR

The paper advances the theory of dendric shifts by providing a general S-adic characterization for minimal dendric shifts over arbitrary alphabets, built from two finite graphs that encode left and right extension data. It proves that minimal dendric shifts are exactly those with primitive -adic representations whose morphisms are dendric return morphisms and that the representation labels infinite paths in both and , yielding concrete criteria for deducing dendricity and eventual dendricity. The framework yields decidability results for eventual dendricity in substitutive settings and provides explicit constructions and examples, including a four-letter case with one right special factor per length and a connection to interval exchanges via planar extension graphs. The work also extends known ternary results to general alphabets and situates the theory within the dynamics of derived shifts, return morphisms, and Rauzy-type induction, suggesting broad avenues for future regular-language descriptions of dendric return morphisms and applications to IET codings.

Abstract

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive -adic representation where the morphisms in are positive tame automorphisms of the free group generated by the alphabet. In this paper we give an -adic characterization of this family by means of two finite graphs. As an application, we are able to decide whether a shift space generated by a uniformly recurrent morphic word is (eventually) dendric.
Paper Structure (18 sections, 44 theorems, 48 equations, 8 figures)

This paper contains 18 sections, 44 theorems, 48 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Words, languages and shift spaces
  4. Extension graphs and dendric shifts
  5. Extension graphs in morphic images
  6. Other graphs characterizing dendricity
  7. Stabilization and eventually dendric shifts
  8. Action of morphisms on $G^L(X)$ and $G^R(X)$
  9. $S$-adic characterization of minimal dendric shifts
  10. $S$-adic representations
  11. Characterization via two graphs
  12. An example: $\mathfrak{S}$-adic characterization of minimal dendric shifts with one right special factor of each length
  13. Characterization of eventually dendric shifts
  14. Decidability of eventual dendricity for minimal substitutive shifts
  15. Interval exchanges
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathfrak{S}$ be a family of dendric return morphisms from $\mathcal{A}^*$ to $\mathcal{A}^*$ and let $X$ be a shift space having an $\mathfrak{S}$-adic representation ${\boldsymbol{\sigma}} = (\sigma_n)_{n \geq 0}$. Then $X$ is minimal dendric if and only if ${\boldsymbol{\sigma}}$ is primitiv

Figures (8)

  • Figure 1: The extension graphs of $\varepsilon$ (on the left), $0$ (in the center) and $1$ (on the right) are trees.
  • Figure 2: Graphs $G_n^L(X)$ for $n \in \{0, \dots, 3\}$.
  • Figure 3: The acyclicity of $\mathcal{E}_X(v)$ is necessary in Proposition \ref{['P:equiv connected']}.
  • Figure 4: Multi-clique $G(\{\{0,1\}, \{1,2,3\}, \{0,3\}\})$
  • Figure 5: Graphs $G = G(\{\{0,1\},\{1,2,3\}\})$ (on the left) and $\beta^L(G)$ (on the right)
  • ...and 3 more figures

Theorems & Definitions (97)

  • Theorem 1
  • Proposition 2: Cassaigne and Nicolas CANT_cassaigne
  • Example 3
  • Definition 4
  • Example 5
  • Proposition 6
  • proof
  • Definition 7
  • Proposition 8
  • proof
  • ...and 87 more