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On the Index of Sturmian Words
 in M. Lothaire, Algebraic combinatorics on Words
, 1999
"... this paper is to present a new proof, with some improvements, of a theorem by Mignosi [21] cited below. Let x be an infinite word, and let F (x) be the sets of its factors (subwords). For w 2 F (x), the index of w in x is the greatest integer d such that w ..."
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Cited by 38 (4 self)
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this paper is to present a new proof, with some improvements, of a theorem by Mignosi [21] cited below. Let x be an infinite word, and let F (x) be the sets of its factors (subwords). For w 2 F (x), the index of w in x is the greatest integer d such that w
Return Words In Sturmian And Episturmian Words
, 2000
"... Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent r ..."
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Cited by 30 (5 self)
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Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has exactly two return words in it. Then, considering episturmian infinite words, which are a natural generalization of Sturmian words, we study the position of the occurrences of any factor in such infinite words and we determinate the return words. At last, we apply these results in order to get a kind of balance property of episturmian words and to calculate the recurrence function of these words.
RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS
, 2002
"... Sturmian words are infinite words over a twoletter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturm ..."
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Cited by 20 (0 self)
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Sturmian words are infinite words over a twoletter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.
Characterizations of finite and infinite episturmian words via lexicographic orderings
 European J. Combin. (in
, 2006
"... 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 character ..."
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Cited by 15 (8 self)
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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 2letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian. Key words: combinatorics on words; lexicographic order; episturmian word; Sturmian word; ArnouxRauzy sequence; balanced word; skew word; episkew word 2000 MSC: 68R15 1
Powers in a class of Astrict standard episturmian words, Theoret
 Comput. Sci
"... 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 studie ..."
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Cited by 10 (8 self)
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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 kbonacci word, a generalization of the Fibonacci word to a kletter alphabet (k≥2).
A characterization of balanced episturmian sequences, Electron
 J. Combin
"... It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a k ..."
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Cited by 10 (1 self)
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It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a kletter alphabet, k ≥ 3, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a kletter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel’s conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed k> 2, there is only one way to cover Z by k Beatty sequences. The problem can be translated to combinatorics on words: for a kletter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies. 1
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
, 2009
"... 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 ..."
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Cited by 9 (7 self)
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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.
REVERSALS AND PALINDROMES IN CONTINUED FRACTIONS
"... Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1. ..."
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Cited by 8 (4 self)
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Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1.