Symbolic coding of linear complexity for generic translations of the torus, using continued fractions - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2020

Symbolic coding of linear complexity for generic translations of the torus, using continued fractions

N. Pytheas Fogg
  • Fonction : Auteur
Camille Noûs
Nicolas Bedaride
Jean François Bertazzon
  • Fonction : Collaborateur
Julien Cassaigne
  • Fonction : Collaborateur
  • PersonId : 954952
Paul Mercat
  • Fonction : Collaborateur
  • PersonId : 975565
Thierry Monteil

Résumé

In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension $2$. More generally, in dimension $d\geq 1$, we study extended measured continued fraction algorithms, which associate to each direction a subshift generated by substitutions, called $S$-adic subshift. We give some conditions which imply the existence, for almost every direction, of a translation of the torus $\mathbb{T}^d$ and a nice generating partition, such that the associated coding is a conjugacy with the subshift.

Dates et versions

hal-03089664 , version 1 (28-12-2020)

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Citer

N. Pytheas Fogg, Camille Noûs, Nicolas Bedaride, Jean François Bertazzon, Julien Cassaigne, et al.. Symbolic coding of linear complexity for generic translations of the torus, using continued fractions. 2020. ⟨hal-03089664⟩
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