Characteristic morphisms of generalized episturmian words

In a recent paper with L. Q. Zamboni, the authors introduced the class of ϑ-episturmian words. An infinite word over A is standard ϑ-episturmian, where ϑ is an involutory antimorphism of A*, if its set of factors is closed under ϑ and its left special factors are prefixes. When ϑ is the reversal operator, one obtains the usual standard episturmian words. In this paper, we introduce and study ϑ-characteristic morphisms, that is, morphisms which map standard episturmian words into standard ϑ-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective. In order to prove this result, we also introduce and study a class of biprefix codes which are overlap-free, i.e., any two code words do not overlap properly, and normal, i.e., no proper suffix (prefix) of any code-word is left (right) special in the code. A further result is that any standard ϑ-episturmian word is a morphic image, by an injective ϑ-characteristic morphism, of a standard episturmian word

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

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

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

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

On z-factorization and c-factorization of standard episturmian words

Ziv-Lempel and Crochemore factorization are two kinds of factorizations of words related to text processing. In this paper, we find these factorizations for standard epiesturmian words. Thus the previously known c-factorization of standard Sturmian words is provided as a special case. Moreover, the two factorizations are compared.Comment: 10 page

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word

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

Palindromic Prefixes and Episturmian Words

Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$ such that $n_{i+1} \leq 2 n_i + 1$ for any $i$, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity $\limsup n_{i+1}/n_i$, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.Comment: 27 pages; many minor changes; to appear in J. Comb. Th. Series