On different generalizations of episturmian words

In this paper we study some classes of infinite words generalizing episturmian words, and analyse the relations occurring among such classes. In each case, the reversal operator R is replaced by an arbitrary involutory antimorphism θ of the free monoid A. In particular, we define the class of θ-words with seed, whose “standard” elements (θ-standard words with seed) are constructed by an iterative θ-palindrome closure process, starting from a finite word u0 called the seed. When the seed is empty, one obtains θ-words; episturmian words are exactly the R-words. One of the main theorems of the paper characterizes θ-words with seed as infinite words closed under θ and having at most one left special factor of each length n ≥ N (where N is some nonnegative integer depending on the word). When N = 0 we call such words θ-episturmian. Further results on the structure of θ-episturmian words are proved. In particular, some relationships between θ-words (with or without seed) and θ-episturmian words are shown. c © 2007 Elsevier B.V. All rights reserved.