Directive words of episturmian words: equivalences and normalization

¹ LaCIM, Université du Québec à Montréal, C.P. 8888, succursale Centre-ville, Montréal, Québec, H3C 3P8, Canada
² The Mathematics Institute, Reykjavík University, Kringlan 1, IS-103 Reykjavík, Iceland; amy.glen@gmail.com
³ Université de Picardie Jules Verne, Laboratoire MIS (Modélisation, Information, Systèmes), 33 rue Saint Leu, 80039 Amiens Cedex 1, France; florence.leve@u-picardie.fr; gwenael.richomme@u-picardie.fr

Received: 26 February 2008
Accepted: 17 October 2008

Abstract

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.