Abstract not found

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...

In this paper we study dynamical properties of a class of uniformly recurrent sequences on a k-letter 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 Arnoux-Rauzy 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 S-adic description of the characteristic sequence similar to that given by Arnoux and Rauzy for k = 2, 3. Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy 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 b-digit expansion is an Arnoux-Rauzy sequence, is transcendental. This yields a class of transcendental numbers of arbitrarily large linear complexity. I

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...

Sturmian words are in nite words over a two-letter alphabet that admit a great number of equivalent de nitions. 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.

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 \Gamma 2 (cf.[4], [3]). We show that aPERb [ fa; bg = St " 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 fa; bg. It is also shown that aPERb[fa; bg = 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 (cf.[1]). From this result one can derive in a different way, by using a theorem of Borel 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; p q of w 2 PER " afa; bg and the slope ae = (jwj b + 1)=(jwj a + 1) of the corresponding Christoffel primitive word awb: If p=q has the development in continued fractions [0; h 1 ; :::; h n\Gamma1 ; h n + 1], then ae has the developme...

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 non-periodic words) for the Fibonacci word. 1

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 pseudo-palindromic infinite words: ω ∈ A N is called a pseudo-palindromic 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 k-periods. A factor u of ω is called a pseudo-palindrome if u = φ(ū). We deduce that if u is a subword of a pseudo-palindromic word ω which is a pseudo-palindrome, then every first return to u is also a pseudo-palindrome. In particular, every pseudo-palindromic infinite word begins in an infinite number of distinct pseudo-palindromes. 1.

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 well-known 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

A string p = p0p1 · · · pn−1 of non-negative integers is a Euclidean string if the string (p0 + 1)p1 · · · (pn−1 − 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + · · · + pn−1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stern-Brocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.