Powers in a class of A-strict standard episturmian words

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the $k$-bonacci word: a generalization of the Fibonacci word to a $k$-letter alphabet ($k\geq2$).Comment: 26 pages; extended version of a paper presented at the 5th International Conference on Words, Montreal, Canada, September 13-17, 200

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

A characterization of fine words over a finite alphabet

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue of Theoretical Computer Science

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense" (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.Comment: 18 pages; to appear in the European Journal of Combinatoric

Extremal properties of (epi)Sturmian sequences and distribution modulo 1

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times

On the critical exponent of generalized Thue-Morse words

Automata, Logic and Semantic

Episturmian words: a survey

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.Comment: 36 pages; major revision: improvements + new material + more reference

Powers in a class of A-strict standard episturmian words

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)