Bidimensional sturmian sequences and substitutions

Abstract

Substitutions are powerful tools to study combinatorial properties of sequences. There exist strong characterizations through substitutions of the Sturmian sequences that are S-adic, substitutive or a fixed-point of a substitution. In this paper, we define a bidimensional version of Sturmian sequences and look for analogous characterizations. We prove in particular that a bidimensional Sturmian sequence is always S-adic and give sufficient conditions under which it is either substitutive or a fixed-point of a substitution. © Springer-Verlag Berlin Heidelberg 2005.