Rich, Sturmian, and trapezoidal words

Rich, Sturmian, and trapezoidal words

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal w...

Return words in Sturmian and episturmian words

Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has...

Sturmian Words: Dynamical Systems and Derivated Words

Sturmian Words, Lyndon Words and Trees

We prove some new combinatorial properties of the set PER of all words w having two periods p and q which are coprimes and such that w = p + q 2 [4,3]. We show that aPERb U {a, b} = St n Lynd, where St is the set of the finite factors of all infinite Sturmian words and Lynd is the set of the Lyndon words on the alphabet {a, b}. It is also shown that aPERb U {a, b} = CP, where CP is the set of C...

Dejean's conjecture and Sturmian words

Dejean conjectured that the repetition threshold of a k-letter alphabet is k/(k−1), k = 3, 4. Values of the repetition threshold for k < 5 were found by Thue, Dejean and Pansiot. Moulin-Ollagnier attacked Dejean’s conjecture for 5 ≤ k ≤ 11. Building on the work of Moulin-Ollagnier, we propose a method for deciding whether a given Sturmian word with quadratic slope confirms the conjecture for a ...