New algebraic and geometric constructs arising from Fibonacci numbers

Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$ C n , it is possible to obtain an infinite (right) word $$W_n$$ W n on the binary alphabet $$\{0,1\}$$ { 0 , 1 } , which we will call the nth Carboncettus word. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $$n=5$$ n = 5 . The fifth Carboncettus word $$W_5$$ W 5 is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000 100 100 010 010 01. Finally, we also define a further word $$W_{\infty }$$ W ∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\infty }$$ W ∞ itself.

Quasi-Sturmian colorings on regular trees

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

Lyndon Words and Christoffel Words

This chapter covers the lexicographical ordering of lower Christoffel words, which is equivalent to the ordering by their slopes (Borel and Laubie). Lower Christoffel words are particular Lyndon words. They are maximum for the lexicographical order among Lyndon words of a given slope (Borel and Laubie). They are, together with the upper Christoffel words, the only unbordered finite Sturmian words (Chuan). They are exactly the Lyndon words which are Sturmian words (Berstel and de Luca). The standard factorization of a lower Christoffel word is obtained by cutting before the smallest lexicographical suffix. Finally, they are exactly the Lyndon words which are equilibrated (Melançon).

Conjugates and Factors

Christoffel words are naturally cyclic objects. They may be defined by the Cayley graphs of finite cyclic groups. They have many characterizations through their conjugation classes; among them, one is obtained using the Burrows–Wheeler transform (Mantaci, Restivo, and Sciortino); another one is due to Pirillo. The sets of their circular factors have many remarkable properties; in particular the number of them of length k is k+1, if k is smaller than the length of the Christoffel word, and it is a characteristic property (Borel and the author), reminiscent of the similar property of Sturmian sequences. A related characterization, similar to that of Droubay and Pirillo for Sturmian sequences, rests on the count of palindromic factors. The set of finite Sturmian words, that is, the set of all factors of all Christoffel words, coincides with the set of balanced words (Dulucq and Gouyou–Beauchamps).

From Christoffel Words to Markoff Numbers

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.