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Episturmian words: a survey

Published online by Cambridge University Press:  06 March 2009

Amy Glen  and
Jacques Justin
Amy Glen
Affiliation:
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; [email protected]
Jacques Justin
Affiliation:
LIAFA, Université Paris Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France; [email protected]
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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.

Keywords

Combinatorics on wordsepisturmian wordsArnoux-Rauzy sequencesSturmian wordsepisturmian morphisms.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 3 , July 2009 , pp. 403 - 442
Copyright
© EDP Sciences, 2009

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