复杂度为$2n+1$的几乎处处平衡序列
Almost everywhere balanced sequences of complexity $2n+1$
摘要：
本文研究了第一作者提出的多维连分数算法下的三元序列。该算法是由两个矩阵定义的，我们证明了它是可测同构的方向序列集$\{1,2\}^\mathbb{N}$上的移位。对于一个给定的集合$\mathcal{C}$的两个替换，我们表明，存在一个$\mathcal{C}$ -进序列的每一个向量的字母频率，或等价地，每一个方向序列。我们证明了它们的因子复杂度至多是$2n+1$，并且是$2n+1$当且仅当字母频率是理性独立的当且仅当$\mathcal{C}$ -adic表示是本原的。在这种情况下，序列是树状的。我们还证明了$μ$ -几乎每个$\mathcal{C}$ -进序列是平衡的，其中$μ$是$\{1,2\}^\mathbb{N}$上的任何移位不变遍历Borel概率测度，给出了圆柱$[12121212]$的正测度。我们还证明了与测度$μ$相关的矩阵上循环的第二李雅普诺夫指数为负。
英文摘要：
We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $μ$-almost every $\mathcal{C}$-adic sequence is balanced, where $μ$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $μ$ is negative.
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