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25
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 37 (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...
A generalization of Sturmian sequences; combinatorial structure and transcendence
 Acta Arith
"... In this paper we study dynamical properties of a class of uniformly recurrent sequences on a kletter alphabet with complexity p(n) = (k − 1)n + 1. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of the (binary) Sturmian sequences of Morse and Hedlund. We ..."
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Cited by 33 (5 self)
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In this paper we study dynamical properties of a class of uniformly recurrent sequences on a kletter alphabet with complexity p(n) = (k − 1)n + 1. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of the (binary) Sturmian sequences of Morse and Hedlund. We give two combinatorial algorithms for constructing characteristic ArnouxRauzy sequences. The first method, which is the central idea of the paper, involves a simple combinatorial algorithm for constructing all bispecial words. This description is new even in the Sturmian case. The second is a Sadic description of the characteristic sequence similar to that given by Arnoux and Rauzy for k = 2, 3. ArnouxRauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every ArnouxRauzy sequence contains arbitrarily large subwords of the form V 2+ɛ and in the Sturmian case arbitrarily large subwords of the form V 3+ɛ. Combined with a recent combinatorial version of Ridout’s Theorem due to S. Ferenczi and C. Mauduit, we prove that an irrational number whose base bdigit expansion is an ArnouxRauzy sequence, is transcendental. This yields a class of transcendental numbers of arbitrarily large linear complexity. I
Forbidden Words in Symbolic Dynamics
, 1999
"... We introduce an equivalence relation ' between functions from N to N. By describing a symbolic dynamical system in terms of forbidden words, we prove that the 'equivalence class of the function that counts the minimal forbidden words of a system is a topological invariant of the system. We show ..."
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Cited by 20 (8 self)
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We introduce an equivalence relation ' between functions from N to N. By describing a symbolic dynamical system in terms of forbidden words, we prove that the 'equivalence class of the function that counts the minimal forbidden words of a system is a topological invariant of the system. We show that the new invariant is independent from previous ones, but it is not characteristic. In the case of soc systems we prove that the 'equivalence of the corresponding functions is a decidable question. As a more special application, we show, by using the new invariant, that two systems associated to Sturmian words having \dierent slope" are not conjugate. Classication: Symbolic Dynamics, Combinatoric on words, Automata and Formal Languages. 1 Introduction In this paper we present a new topological invariant for Symbolic Dynamics. The techniques we use and some complementary results are from Combinatorics on words and from the theory of Automata and Formal Languages. Indeed there...
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.
Sturmian words, Lyndon words and trees
, 1997
"... We prove some new combinatorial properties of the set PER of all words w having two periods p and q which are coprimes and such that w = p + q 2 [4,3]. We show that aPERb U {a, b} = St n Lynd, where St is the set of the finite factors of all infinite Sturmian words and Lynd is the set of the Lyndon ..."
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Cited by 16 (1 self)
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We prove some new combinatorial properties of the set PER of all words w having two periods p and q which are coprimes and such that w = p + q 2 [4,3]. We show that aPERb U {a, b} = St n Lynd, where St is the set of the finite factors of all infinite Sturmian words and Lynd is the set of the Lyndon words on the alphabet {a, b}. It is also shown that aPERb U {a, b} = CP, where CP is the set of Christoffel primitive words. Such words can be defined in terms of the ‘slope ’ of the words and of their prefixes [l]. From this result one can derive in a different way, by using a theorem of Bore1 and Laubie, that the elements of the set aPERb are Lyndon words. We prove the following correspondence between the ratio p/q of the periods p,q, pdq of w E PER f ~ a(a, b} * and the slope p = (lw(h + I)/ ( Iwl. + 1) of the corresponding Christoffel primitive word awb: If p/q has the development in continued fractions [0, hl,...,!I, _ 1, h, + 11, then p has the development in continued fractions [0, h,,..., hi, hl + 11. This and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the StemBrocot and the Raney numbers of a node in a complete binary tree. However, one needs some further results. More precisely we label the binary tree with standard pairs (standard tree), Christoffel pairs (Christoffel tree) and the elements of PER (Farey tree). The main theorem is the following: If the node W is labeled by the standard pair (u, o), by the Christoffel pair (n,~) and by w E PER, then uv = wab, xy = awb. The StemBrocot number SB ( W) is equal to the slope of the standard word uv and of the Christoffel word xy while the Raney number Ra ( W) is equal to the ratio of the minimal period of wa and the minimal period of wb. Some further auxiliary results are also derived.
Palindrome prefixes and episturmian words”, preprint math.CO/0501420, submitted
"... Abstract: Let w be an infinite word on an alphabet A. We denote by (ni)i≥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 ni+1 ≤ 2ni + 1 for all i, and study these words. Special exa ..."
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Cited by 13 (1 self)
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Abstract: Let w be an infinite word on an alphabet A. We denote by (ni)i≥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 ni+1 ≤ 2ni + 1 for all 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 lim sup ni+1/ni, and prove that it is minimal (among all nonperiodic words) for the Fibonacci word. 1
Palindromes and pseudo palindromes in episturmian and pseudoepisturmian infinite words
 in: S. Brlek, C. Reutenauer (Eds.), Words 2005, n. 36 in Publications du LACIM, 2005
"... ABSTRACT. Let A be a finite set of cardinality greater or equal to 2. An infinite word ω ∈ A N is called Episturmian if it is closed under mirror image (meaning if u = u1u2 · · · uk is a subword of ω, then so is ū = uk · · · u2u1) and if for every n ≥ 1 there exists at most one subword u of ω of ..."
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Cited by 10 (6 self)
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ABSTRACT. Let A be a finite set of cardinality greater or equal to 2. An infinite word ω ∈ A N is called Episturmian if it is closed under mirror image (meaning if u = u1u2 · · · uk is a subword of ω, then so is ū = uk · · · u2u1) and if for every n ≥ 1 there exists at most one subword u of ω of length n which is right special. We show that if u is a subword of an Episturmian word ω which is a palindrome, then every first return to u is also a palindrome. As a consequence, every Episturmian word begins in an infinite number of distinct palindromes. Our methods extend to the context of pseudopalindromic infinite words: ω ∈ A N is called a pseudopalindromic word if there exists a bijection φ: A → A with φ 2 the identity such that for each subword u of ω we have that φ(ū) is also a subword of ω, and for every n ≥ 1 there exists at most one subword u of ω of length n which is right special. These words arise naturally in the context of the Fine and Wilf Theorem on kperiods. A factor u of ω is called a pseudopalindrome if u = φ(ū). We deduce that if u is a subword of a pseudopalindromic word ω which is a pseudopalindrome, then every first return to u is also a pseudopalindrome. In particular, every pseudopalindromic infinite word begins in an infinite number of distinct pseudopalindromes. 1.
Sturmian morphisms, the braid group B4, Christoffel words and bases of F2
, 2005
"... Abstract. We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F2) of automorphisms of the rank two free group F2 and show that it can be realized as a monoid in the group B4 of braids on four strings. In the second part we use Ch ..."
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Cited by 8 (2 self)
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Abstract. We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F2) of automorphisms of the rank two free group F2 and show that it can be realized as a monoid in the group B4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F2 lifting any given basis of the free abelian group Z 2. We further give an algorithm allowing to decide whether two elements of F2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.
On Sturmian and episturmian words, and related topics
, 2006
"... Combinatorics on words plays a fundamental role in various fields of mathematics, not to mention its relevance in theoretical computer science and physics. Most renowned among its branches is the theory of infinite binary sequences called Sturmian words, which are fascinating in many respects, havin ..."
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Cited by 6 (5 self)
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Combinatorics on words plays a fundamental role in various fields of mathematics, not to mention its relevance in theoretical computer science and physics. Most renowned among its branches is the theory of infinite binary sequences called Sturmian words, which are fascinating in many respects, having been studied from combinatorial, algebraic, and geometric points of view. The most wellknown example of a Sturmian word is the ubiquitous Fibonacci word, the importance of which lies in combinatorial pattern matching and the theory of words. Properties of the Fibonacci word and, more generally, Sturmian words have been extensively studied, not only because of their significance in discrete mathematics, but also due to their practical applications in computer imagery (digital straightness), theoretical physics (quasicrystal modelling) and molecular biology. The history of Sturmian words dates back to the astronomer J. Bernoulli III (1772) and, as described in Venkov’s book [38], there also exists some early work by Christoffel (1875) and Markoff (1882). The first detailed investigation of Sturmian words was carried out in 1940 by Morse and Hedlund [33], who studied such words under the framework of symbolic dynamics and, in fact, introduced the term “Sturmian”, named after the mathematician Charles François