Towards a statement of the S-adic conjecture through examples

Fabien Durand (1) , Julien Leroy (1) , Gwenaël Richomme (2, 3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 ARITH - Arithmétique informatique
3 UPVM - Université Paul-Valéry - Montpellier 3

The $S$-adic conjecture claims that there exists a condition $C$ such that a sequence has a sub-linear complexity if and only if it is an $S$-adic sequence satisfying Condition $C$ for some finite set $S$ of morphisms. We present an overview of the factor complexity of $S$-adic sequences and we give some examples that either illustrate some interesting properties or that are counter-examples to what could be believed to be ''a good Condition $C$''.

Mots clés: S-adic conjecture; combinatorics on words; substitution; word complexity; morphisms

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Fabien Durand, Julien Leroy, Gwenaël Richomme. Towards a statement of the S-adic conjecture through examples. 2012. ⟨hal-00724788⟩