By Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco V. Saliola
The 2 elements of this article are in keeping with sequence of lectures introduced by means of Jean Berstel and Christophe Reutenauer in March 2007 on the Centre de Recherches Mathematiques, Montreal, Canada. half I represents the 1st smooth and finished exposition of the speculation of Christoffel phrases. half II offers a number of combinatorial and algorithmic elements of repetition-free phrases stemming from the paintings of Axel Thue--a pioneer within the concept of combinatorics on phrases. A newbie to the idea of combinatorics on phrases can be prompted by way of the varied examples, and the big number of routines, which make the booklet precise at this point of exposition. The fresh and streamlined exposition and the huge bibliography can also be liked. After examining this booklet, newbies may be able to learn smooth examine papers during this quickly growing to be box and give a contribution their very own study to its improvement. skilled readers may be drawn to the finitary method of Sturmian phrases that Christoffel phrases supply, in addition to the radical geometric and algebraic strategy selected for his or her exposition. they're going to additionally enjoy the ancient presentation of the Thue-Morse observe and its purposes, and the unconventional effects on Abelian repetition-free phrases.
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Extra resources for Combinatorics on words: Christoffel words and repetitions in words
44 CHAPTER 5. PRIMITIVE ELEMENTS IN THE FREE GROUP F2 Proof. Suppose |ad − bc| = 1. Then (1, 0) and (0, 1) are in the span of (a, b) and (c, d). Indeed, d(a, b) − b(c, d) = (ad − bc, 0) = ±(1, 0) and a(c, d) − c(a, b) = ±(0, 1). Thus (a, b) and (c, d) generate Z2 . Conversely, suppose (a, b) and (c, d) generate Z2 . Then the matrix equation a c x=b b d has a unique solution x ∈ Z2 for all vectors b ∈ Z2 . For b = (0, 1)T , we have x= a c b d −1 0 1 = 1 ad − bc d −c −b a 0 1 = 1 ad − bc −c a ∈ Z2 .
4 (Berstel, de Luca [BdL1997]). A word w is a Christoffel word if and only if it is a balanced1 Lyndon word. 3 Balanced2 Lyndon words There is another notion of “balanced” that may be given to Lyndon words. Before defining it, we need to introduce a fundamental result about Lyndon words. Recall that a word w is a Lyndon word if and only if for all nontrivial factorizations w = (u, v), w < vu in the lexicographic order. Note that we did not allow w ≤ vu. Otherwise stated, we demand that Lyndon words are primitive.
This led Guy Melan¸con [Mel1999] to introduce a second, recursive, notion of balanced: call a Lyndon word w balanced2 if w is a letter or there is a factorization w = (u, v) that is both a left and right factorization with the further property that u and v are balanced2 . Example. The reader may check that aabaacab, xxyy, and xxyxxyxy are Lyndon words. Among these, only xxyxxyxy is balanced2 . 4. 4: The left and right factorizations of three Lyndon words. Only xxyxxyxy, the Christoffel word of slope 35 , is seen to be a balanced2 Lyndon word.