Episturmian words: a survey

Issue		RAIRO-Theor. Inf. Appl. Volume 43, Number 3, July-September 2009


Page(s)		403 - 442
DOI		10.1051/ita/2009003
Published online		06 March 2009

RAIRO-Theor. Inf. Appl. 43, 403-442 (2009)
DOI: 10.1051/ita/2009003

Amy Glen¹ and Jacques Justin²

¹ LaCIM, Université du Québec à Montréal, C.P. 8888, succursale Centre-ville, Montréal, Québec, H3C 3P8, Canada; The Mathematics Institute, Reykjavik University, Kringlan 1, IS-103 Reykjavik, Iceland; amy.glen@gmail.com
² LIAFA, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France; jacjustin@free.fr

Received December 11, 2007. Accepted December 4, 2008. Published online 6 March 2009

Abstract
In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and “episkew words” that generalize the skew words of Morse and Hedlund.

Mathematics Subject Classification. 68R15.

Key words: Combinatorics on words -- episturmian words -- Arnoux-Rauzy sequences -- Sturmian words -- episturmian morphisms.

© EDP Sciences 2009

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