Abstract

We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to $\mathbb{Z}^3$. In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.

Highlights

  • À l’algorithme de fraction continue soustractif décrit par l’itération de l’application R+ 2 → R+ 2(x, y) → (x − y, y) si x ≥ y,(x, y − x) sinon, est associée une classe particulière de mots infinis binaires appelés mots sturmiens

  • The abelianized vector of u, sometimes called Parikh vector of u, is the vector ab(u) = (|u|α)α∈A, which counts the number of times that each letter occurs in the finite word u

  • We propose to show that the word w∞ satisfies a stronger property: its Rauzy fractal is unbounded in all directions of the plane

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Summary

Introduction

À l’algorithme de fraction continue soustractif décrit par l’itération de l’application (dite de Farey) R+ 2 → R+ 2. Le théorème d’Oseledets [8] suggère toutefois que ces fractals sont contenus dans une bande du plan ; en effet, si les exposants de Lyapounov associés au produit de matrices donné par l’algorithme existent, l’un de ces exposants au moins doit être négatif puisque leur somme est nulle. La construction que nous présentons s’adapte à la classe des mots associée à l’algorithme de fraction continue multidimensionnelle de Cassaigne–Selmer, introduite dans [5], ainsi qu’aux mots épisturmiens stricts, qui sont la généralisation des mots d’Arnoux–Rauzy. The vector of letter frequencies of any Arnoux–Rauzy word has rationally independent entries This theorem completes the works of Arnoux and Starosta, who conjectured it in 2013, to prove that the Arnoux–Rauzy continued fraction algorithm detects all kind of rational dependencies [3]. The vector of letter frequencies of any strict episturmian word over {1, . . . , d } has rationally independent entries

Preliminaries
Results
Proof of Lemma 3

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