Syndeticity and independent substitutions

Fabien Durand (1) , Michel Rigo (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 Département de Mathématiques [Liège]

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence $x$ is generated by two independent substitutions, at least one being of exponential growth, then the factors of $x$ appearing infinitely often in $x$ appear with bounded gaps. As an application, we derive an analogue of Cobham's theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential.

Mots clés: Cobham; système de numération; substitution

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

Fabien Durand, Michel Rigo. Syndeticity and independent substitutions. Advances in Applied Mathematics, 2009, 42, pp.1-22. ⟨hal-00407667⟩