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Powers in a class of A-strict standard episturmian words

Glen, A.ORCID: 0000-0002-9434-3412 (2005) Powers in a class of A-strict standard episturmian words. In: 5th International Conference on Words, 13 - 17 September, Quebec, Canada

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Introduced by Droubay, Justin and Pirillo [8], episturmian words are a natural extension of the well-known family of Sturmian words (aperiodic infinite words of minimal complexity) to an arbitrary finite alphabet. In this paper, the study of episturmian words is continued in more detail. In particular, for a specific class of episturmian words (a typical element of which we shall denote by s), we will explicitly determine all the integer powers occurring in its constituents. This has recently been done in [6] for Sturmian words, which are exactly the aperiodic episturmian words over a two-letter alphabet.

A finite word w is said to have an integer power in an infinite word x if wp = ww • • •w (p times) is a factor of x for some integer p _ 2. Here, our analysis of powers occurring in episturmian words s hinges on canonical decompositions in terms of their ‘building blocks’. Another key tool is a generalization of singular words, which were first defined in [17] for the ubiquitous Fibonacci word, and later extended to Sturmian words in [15] and the Tribonacci sequence in [16]. Our generalized singular words will prove to be useful in the study of factors of episturmian words, just as they have for Sturmian words.

This paper is organized as follows. After some preliminaries (Section 2), we define, in Section 3, a restricted class of episturmian words upon which we will focus for the rest of the paper. A typical element of this class will be denoted by s. In Section 4, we give some simple results which, in turn, lead us to a generalization of singular words for episturmian words s. The index, i.e., maximal fractional power, of the building blocks of s is then studied in Section 5. Finally, in Section 6, we determine all squares (and subsequently higher powers) occurring in s. The main results are demonstrated via the k-bonacci word; a generalization of the Fibonacci word to a k-letter alphabet (k _ 2).

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