Gwénaël Richomme

Abelian complexity of minimal subshifts

In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerdens theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words, we prove something stronger: for every Sturmian word ω and positive integer k, each sufficiently long factor of ω begins with an Abelian k-power.

Some characterizations of Parikh matrix equivalent binary words

Abstract We state different characterizations of pair of words having the same Parikh matrix.

Conjugacy and episturmian morphisms

Episturmian morphisms generalize Sturmian morphisms. Here, we study some intrinsic properties of these morphisms: invertibility, presentation, cancellativity, unitarity, characterization by conjugacy. Most of them are generalizations of known properties of Sturmian morphisms. But we present also some results on episturmian morphisms that have not already been stated in the particular case of Sturmian morphisms: characterization of the episturmian morphisms that preserve palindromes, new algorithms to compute conjugates.We also study the conjugation of morphisms in the general case and show that the monoid of invertible morphisms on an alphabet containing at least three letters is not finitely generated.

AVOIDING ABELIAN POWERS IN BINARY WORDS WITH BOUNDED ABELIAN COMPLEXITY

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerdens theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo–Varricchio and Halbeisen–Hungerbuhler.

Some results on k -power-free morphisms

One way to generate infinite k-power-free words is to iterate a k-power-free morphism, that is a morphism that preserves finite k-power-free words. We first prove that the monoid of k-power-free endomorphisms on an alphabet containing at least three letters is not finitely generated. Test-sets for k-power-free morphisms (that is, the sets T such that a morphism f is k-power-free if and only if f(T) is k-power-free) give characterizations of these morphisms. In the case of binary morphisms and k=3, we prove that a set T of cube-free words is a test-set for cube-freeness if and only if it contains 12 particular factors. Consequently, a morphism f on {a,b} is cube-free if and only if f(aabbababbabbaabaababaabb) is cube-free (length 24 is optimal). Another consequence is an unpublished result of Leconte: a binary morphism is cube-free if and only if the images of all cube-free words of length 7 are cube-free. When k3, we show that no finite test-set exists for morphisms defined on an alphabet containing at least three letters. In the last part, we show that to generate an infinite cube-free word by iterating a morphism, we do not necessarily need a cube-free morphism. We give a new characterization of some morphisms that generate infinite cube-free words.

Quasiperiodic Sturmian words and morphisms

We characterize all quasiperiodic Sturmian words: A Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity.

Characterization of test-sets for overlap-free morphisms

We give a characterization of all the sets X such that any morphism h on {a, b} is overlap-free if and only if for all x in X, h(x) is overlap-free. As a consequence, we observe the particular case X = {bbabaa} which improves the previous characterization of Berstel-Seebold.

Conjugacy of morphisms and Lyndon decomposition of standard Sturmian words

Using the notions of conjugacy of morphisms and of morphisms preserving Lyndon words, we answer a question of G. Melancon. We characterize cases where the sequence of Lyndon words in the Lyndon factorization of a standard Sturmian word is morphic. In each possible case, the corresponding morphism is given.

On a conjecture about finite fixed points of morphisms

A conjecture of M. Billaud is: given a word w, if, for each letter x occurring in w, the word obtained by erasing all the occurrences of x in w is a fixed point of a nontrivial morphism fx, then w is also a fixed point of a non-trivial morphism. We prove that this conjecture is equivalent to a similar one on sets of words. Using this equivalence, we solve these conjectures in the particular case where each morphism fx has only one expansive letter.

Quasiperiodic and Lyndon episturmian words

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if, it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.