论克里斯托弗与标准词及其派生词
On Christoffel and standard words and their derivatives
摘要：
我们介绍和研究自然衍生物Christoffel和有限的标准字，以及特征Sturmian字。这些衍生物，这是实现为逆图像下适当的态射，保持上述类别的话。在Christoffel词的情况下，所涉及的态射将$a$映射到$a^{k+1}b$（分别为，~ $ab^{k}$）和$b$至$a^{k}b$（分别为~ $ab^{k+1}$），以获得合适的$k>0$。只要导数的长度大于一个字母，自然就会得到更高阶的导数。我们将Christoffel或标准词的深度定义为导数为单个字母的最小阶。我们给出了几个组合和算术描述的深度，和（紧）下限和上限。
英文摘要：
We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.
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