论文标题
使用持续分数
Symbolic coding of linear complexity for generic translations of the torus, using continued fractions
论文作者
论文摘要
在本文中，我们证明了$ \ mathbb {t}^2 $的几乎每个翻译都允许具有线性复杂性$ 2N+1 $的符号编码。该分区是由与替代序列相关的Rauzy分形构造的，这些分形是由特定的延长持续持续分数算法在投影尺寸$ 2 $中产生的。更普遍的是，在尺寸$ d \ geq 1 $中，我们研究了扩展测量的持续分数算法，该算法与每个方向相关联，由替换产生的子班置换档，称为$ s $ adic subshift。我们提供了一些条件，这些条件暗示了几乎每个方向的存在，即圆环$ \ mathbb {t}^d $的翻译和一个不错的生成分区，因此相关的编码是与亚转移的共轭。
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.
