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

Abstract

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 $d\ge 1$, 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.

Cited by in corpus (1)

• Geometrical representation of subshifts for primitive substitutions