By Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco V. Saliola

The 2 components of this article are in response to sequence of lectures introduced by way of Jean Berstel and Christophe Reutenauer in March 2007 on the Centre de Recherches Mathematiques, Montreal, Canada. half I represents the 1st glossy and finished exposition of the speculation of Christoffel phrases. half II offers quite a few combinatorial and algorithmic elements of repetition-free phrases stemming from the paintings of Axel Thue--a pioneer within the thought of combinatorics on phrases. A newbie to the idea of combinatorics on phrases should be prompted by means of the varied examples, and the big number of routines, which make the booklet specified at this point of exposition. The fresh and streamlined exposition and the huge bibliography can also be favored. After interpreting this publication, newcomers can be able to learn glossy study 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 technique selected for his or her exposition. they are going to additionally have fun with the historic presentation of the Thue-Morse be aware and its functions, and the radical effects on Abelian repetition-free phrases.

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**Sample text**

Suppose g ∈ F2 is of length r and suppose that u and v are positive elements with v = gug−1 . Let g = a1 · · · ar be a reduced expression for g, where ai ∈ {x, y, x−1 , y −1 } for 1 ≤ i ≤ r. We consider three cases. (i): If ℓ(gu) < ℓ(g) + ℓ(u), then the first letter z of u must be a−1 r ∈ {x, y}. Write u = zu1 and g = hz −1 for some u1 ∈ {x, y}∗ and some h ∈ F2 . Then v = gug−1 = (hz −1 )(zu1 )(hz −1 )−1 = h(u1 z)h−1 . Since ℓ(h) < r and z ∈ {x, y}, it follows from the induction hypothesis that u1 z and v are conjugate in {x, y}∗ .

If ǫ = p1 , p2 , . . , pr denote the sequence of palindromic prefixes of Pal(w) different than Pal(w) listed in order of increasing length, and if z1 , z2 , . . , zr ∈ {x, y} denote the letters in Pal(w) immediately following the prefixes p1 , p2 , . . , pr in Pal(w), then w = z1 z2 · · · zr . 10. If w = vzu, where u does not have an occurrence of the letter z, then Pal(wz) = Pal(w) Pal(v)−1 Pal(w), where the product is evaluated in the free group generated by {x, y}. 11. 7) Let αx = G = (x, xy) and αy = D = (yx, y).

3. The third is explained by our next characterization of Christoffel words. 2 (Mantaci, Restivo, Sciortino [MRS2003]). A word w ∈ {x, y}∗ is a conjugate of a Christoffel word if and only if BW T (w) takes the form y q xp , where p ⊥ q. 1. 1. 1. 2. The Burrows–Wheeler transform BW T is injective on Lyndon words (cf. [CDP2005] for more details): Given a Lyndon word w = a1 a2 · · · an , let b1 b2 · · · bn and c1 c2 · · · cn denote the first and last columns, respectively, of the Burrows–Wheeler matrix.