论文标题
与$ n $连接的分数算法相关的Sturmian序列的概括
Generalizations of Sturmian sequences associated with $N$-continued fraction algorithms
论文作者
论文摘要
鉴于零与一个之间的正整数$ n $和$ x $不合理，因此$ n $连接的分数扩展为$ x $，类似于经典的持续分数扩展，但分子都等于$ n $。受Sturmian序列的启发，我们介绍了$ n $连接的分数序列$ω（x，n）$和$ \hatΩ（x，n）$，这与$ n $连接的分数扩展相关。它们是在两个字母字母上获得的无限单词，该字母是某些替代的指令序列的极限，因此它们是$ s $ adic序列。当$ n = 1 $时，我们是经典的持续分数算法的情况，并获得了众所周知的sturmian序列。我们表明$ω（x，n）$和$ \hatΩ（x，n）$是$ c $的，对于某些明确的$ c $，并计算其因子复杂性函数。我们还获得了统一的单词频率并推断出相关的子迁移的独特登山性。最后，我们为$ n $ contin的分数扩展提供了类似Farey的地图，该地图提供了$ n $连接的分数的添加版本，我们证明了这种分数，并明确提供了不变的度量。
Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired by Sturmian sequences, we introduce the $N$-continued fraction sequences $ω(x,N)$ and $\hatω(x,N)$, which are related to the $N$-continued fraction expansion of $x$. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are $S$-adic sequences. When $N=1$, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that $ω(x,N)$ and $\hatω(x,N)$ are $C$-balanced for some explicit values of $C$ and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for $N$-continued fraction expansions, which provides an additive version of $N$-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.
