Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne; Sébastien Ferenczi; Luca Q. Zamboni

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1265-1276
  • ISSN: 0373-0956

Abstract

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In a 1982 paper Rauzy showed that the subshift ( X , T ) generated by the morphism 1 ↦ 12 , 2 ↦ 13 and 3 ↦ 1 is a natural coding of a rotation on the two-dimensional torus 𝕋 2 , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ℝ 2 , each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity 2 n + 1 satisfying a combinatorial criterion known as the ☆ condition of Arnoux and Rauzy codes the orbit of a point under a rotation on 𝕋 2 . In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence ω * which is unbalanced in the following sense: for each N > 0 there exist two factors of ω * of equal length, with one having at least N more occurrences of a given letter than the other. We then invoke a result due to Rauzy on bounded remainder sets to establish the existence of an Arnoux-Rauzy sequence which is not a natural coding of a rotation on 𝕋 2 .

How to cite

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Cassaigne, Julien, Ferenczi, Sébastien, and Zamboni, Luca Q.. "Imbalances in Arnoux-Rauzy sequences." Annales de l'institut Fourier 50.4 (2000): 1265-1276. <http://eudml.org/doc/75456>.

@article{Cassaigne2000,
abstract = {In a 1982 paper Rauzy showed that the subshift $(X,T)$ generated by the morphism $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$ is a natural coding of a rotation on the two-dimensional torus $\{\Bbb T\}^2$, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $\{\Bbb R\}^2,$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2n+1$ satisfying a combinatorial criterion known as the $\star $ condition of Arnoux and Rauzy codes the orbit of a point under a rotation on $\{\Bbb T\}^2$. In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence $\omega _* $ which is unbalanced in the following sense: for each $N&gt;0$ there exist two factors of $\omega _* $ of equal length, with one having at least $N$ more occurrences of a given letter than the other. We then invoke a result due to Rauzy on bounded remainder sets to establish the existence of an Arnoux-Rauzy sequence which is not a natural coding of a rotation on $\{\Bbb T\}^2$.},
author = {Cassaigne, Julien, Ferenczi, Sébastien, Zamboni, Luca Q.},
journal = {Annales de l'institut Fourier},
keywords = {infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences},
language = {eng},
number = {4},
pages = {1265-1276},
publisher = {Association des Annales de l'Institut Fourier},
title = {Imbalances in Arnoux-Rauzy sequences},
url = {http://eudml.org/doc/75456},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Ferenczi, Sébastien
AU - Zamboni, Luca Q.
TI - Imbalances in Arnoux-Rauzy sequences
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1265
EP - 1276
AB - In a 1982 paper Rauzy showed that the subshift $(X,T)$ generated by the morphism $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$ is a natural coding of a rotation on the two-dimensional torus ${\Bbb T}^2$, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ${\Bbb R}^2,$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2n+1$ satisfying a combinatorial criterion known as the $\star $ condition of Arnoux and Rauzy codes the orbit of a point under a rotation on ${\Bbb T}^2$. In this note we exhibit a counterexample to this conjecture. We first build an Arnoux-Rauzy sequence $\omega _* $ which is unbalanced in the following sense: for each $N&gt;0$ there exist two factors of $\omega _* $ of equal length, with one having at least $N$ more occurrences of a given letter than the other. We then invoke a result due to Rauzy on bounded remainder sets to establish the existence of an Arnoux-Rauzy sequence which is not a natural coding of a rotation on ${\Bbb T}^2$.
LA - eng
KW - infinite words; codings of rotations; return times; bounded reaminder sets; balanced sequences; Arnoux-Rauzy sequences; Sturmian sequences
UR - http://eudml.org/doc/75456
ER -

References

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Citations in EuDML Documents

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  1. Christiane Frougny, Zuzana Masáková, Edita Pelantová, Complexity of infinite words associated with beta-expansions
  2. Christiane Frougny, Zuzana Masáková, Edita Pelantová, Complexity of infinite words associated with beta-expansions
  3. Julien Cassaigne, Sébastien Ferenczi, Ali Messaoudi, Weak mixing and eigenvalues for Arnoux-Rauzy sequences
  4. Ondřej Turek, Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
  5. Boris Adamczewski, Codages de rotations et phénomènes d'autosimilarité
  6. Valérie Berthé, Timo Jolivet, Anne Siegel, Connectedness of fractals associated with Arnoux–Rauzy substitutions
  7. Julien Bernat, Study of irreducible balanced pairs for substitutive languages
  8. Sébastien Ferenczi, Charles Holton, Luca Q. Zamboni, Structure of three interval exchange transformations I: an arithmetic study
  9. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets
  10. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets

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