RAIRO - Theoretical Informatics and Applications

Research Article

Episturmian words: a survey

Amy Glena1 and Jacques Justina2

a1 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

a2 LIAFA, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France; jacjustin@free.fr


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.

(Received December 11 2007)

(Accepted December 4 2008)

(Online publication March 6 2009)

Key Words:

  • Combinatorics on words;
  • episturmian words;
  • Arnoux-Rauzy sequences;
  • Sturmian words;
  • episturmian morphisms.

Mathematics Subject Classification:

  • 68R15