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

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.