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34
Recent Results In Sturmian Words
, 1996
"... In this survey paper, we present some recent results concerning finite and infinite Sturmian words. We emphasize on the different definitions of Sturmian words, and various subclasses, and give the ways to construct them related to continued fraction expansion. Next, we give properties of special ..."
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Cited by 36 (2 self)
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In this survey paper, we present some recent results concerning finite and infinite Sturmian words. We emphasize on the different definitions of Sturmian words, and various subclasses, and give the ways to construct them related to continued fraction expansion. Next, we give properties of special finite Sturmian words, called standard words. Among these, a decomposition into palindromes, a relation with the periodicity theorem of Fine and Wilf, and the fact that all these words are Lyndon words. Finally, we describe the structure of Sturmian morphisms (i.e. morphisms that preserve Sturmian words) which is now rather well understood. 1 Introduction Combinatorial properties of finite and infinite words are of increasing importance in various fields of physics, biology, mathematics and computer science. Infinite words generated by various devices have been considered [9]. We are interested here in a special family of infinite words, namely Sturmian words. Sturmian words represent...
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.
A characterization of Sturmian morphisms
 Mathematical Foundations of Computer Science 1993, Lecture Notes in Computer Science
, 1993
"... Abstract. A morphism is called Sturmian if it preserves all Sturmian (infinite) words. It is weakly Sturmian if it preserves at least one Sturmian word. We prove that a morphism is Sturmian if and only if it keeps the word ba2ba2baba 2 bab balanced. As a consequence, weakly Sturmian morphisms are St ..."
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Cited by 16 (1 self)
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Abstract. A morphism is called Sturmian if it preserves all Sturmian (infinite) words. It is weakly Sturmian if it preserves at least one Sturmian word. We prove that a morphism is Sturmian if and only if it keeps the word ba2ba2baba 2 bab balanced. As a consequence, weakly Sturmian morphisms are Sturmian. An application to infinite words associated to irrational numbers is given. 1
Hierarchical Structures in Sturmian Dynamical Systems
"... The paper is concerned with hierarchical structures in subshifts over a nite alphabet. In particular, we present a hierarchy based approach to Sturmian systems. This approach is then used to characterize the linearly repetitive Sturmian systems (among the Sturmian systems) by uniform positivity ..."
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Cited by 13 (12 self)
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The paper is concerned with hierarchical structures in subshifts over a nite alphabet. In particular, we present a hierarchy based approach to Sturmian systems. This approach is then used to characterize the linearly repetitive Sturmian systems (among the Sturmian systems) by uniform positivity of certain weights. More generally, we discuss various bounds on weights and their relationship.
Sequences Of Low Complexity: Automatic And Sturmian Sequences
"... The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences ..."
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Cited by 6 (0 self)
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The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences. We discuss their topological and measuretheoretic properties, by introducing some classical tools in combinatorics on words and in the study of symbolic dynamical systems. 1 Introduction The aim of this course is to introduce two characteristic families of sequences of low "complexity": automatic sequences and Sturmian sequences (complexity is defined here as the combinatorial function which counts the number of factors of given length of a sequence over a finite alphabet). These sequences not only occur in many mathematical fields but also in various domains as theoretical computer science, biology, physics, cristallography... We first define some classical tools in combinatorics on...
Frequencylocking on the Boundary of the Barycentre Set
 Experimental Mathematics
"... . We consider the doubling map T : z 7! z 2 of the circle. For each T invariant probability measure ¯ we define its barycentre b(¯) = R S 1 zd¯(z), which describes its average weight around the circle. We study the set\Omega of all such barycentres, a compact convex set with nonempty interior ..."
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Cited by 4 (2 self)
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. We consider the doubling map T : z 7! z 2 of the circle. For each T invariant probability measure ¯ we define its barycentre b(¯) = R S 1 zd¯(z), which describes its average weight around the circle. We study the set\Omega of all such barycentres, a compact convex set with nonempty interior. We obtain a numerical approximation of the boundary @\Omega . This appears to have a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequencylocking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit. 1991 Mathematics Subject Classification: 58F11, 58F15, 58F03 Section 1. Introduction. A recurring theme in the study of nonlinear dynamics is the occurrence of nonsmooth phenomena (irregular conjugacies, Cantorlike attractors, intricate Julia sets, fractal descriptions of bifurcations, etc), even when the syste...
Conjugates of characteristic Sturmian words generated by morphisms
 European J. Combin
"... This article is concerned with characteristic Sturmian words of slope α and 1 −α (denoted by cα and c1−α respectively), where α ∈ (0, 1) is an irrational number such that α = [0; 1 + d1, d2,..., dn] with dn ≥ d1 ≥ 1. It is known that both cα and c1−α are fixed points of nontrivial (standard) morphi ..."
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Cited by 3 (3 self)
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This article is concerned with characteristic Sturmian words of slope α and 1 −α (denoted by cα and c1−α respectively), where α ∈ (0, 1) is an irrational number such that α = [0; 1 + d1, d2,..., dn] with dn ≥ d1 ≥ 1. It is known that both cα and c1−α are fixed points of nontrivial (standard) morphisms σ and ˆσ, respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words cα and c1−α are generated by the respective morphisms σ and ˆσ. For the particular case when α = [0; 2, r] (r ≥ 1), we give a decomposition of each conjugate of cα (and hence c1−α) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word.
Repetitive Perhaps, But Certainly Not Boring
"... In this paper some of the work done on repetitions in strings is surveyed, especially that of an algorithmic nature. Several open problems are described and conjectures formulated about some of them. KEYWORDS: string, word, repetition, repeat, cover 1 INTRODUCTION Repetitions in strings are usual ..."
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In this paper some of the work done on repetitions in strings is surveyed, especially that of an algorithmic nature. Several open problems are described and conjectures formulated about some of them. KEYWORDS: string, word, repetition, repeat, cover 1 INTRODUCTION Repetitions in strings are usually thought of as adjacent or \tandem"; that is, the string uvu is counted as a repetition of u if and only if v = , the empty string. However, in certain contexts  for example, DNA sequence analysis [S98], data compression [IS98], analysis of musical texts [CIR96]  this denition may be too narrow. Here therefore we take a wider view and regard uvu as a repetition of a nonempty string u for any nite string v. Even more generally, we also count as repetitions cases where the string u overlaps itself; for example, abaabaab is accepted as a repetition of abaab. In order to make sure that these ideas are clear, we express them more formally. Throughout this paper x will denote a string of l...