Sturmian words and words with a critical exponent

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...

A note on Sturmian words

We describe an algorithm which, given a factor of a Sturmian word, computes the next factor of the same length in the lexicographic order in linear time. It is based on a combinatorial property of Sturmian words which is related with the Burrows-Wheeler transformation.

The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n− 2)/(2n− 2), and this exponent is attained at position 1.