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...

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.

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.

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

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 measure-theoretic 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...

. 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 non-empty interior. We obtain a numerical approximation of the boundary @\Omega . This appears to have a countable dense set of points of non-differentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency-locking 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 non-linear dynamics is the occurrence of non-smooth phenomena (irregular conjugacies, Cantor-like attractors, intricate Julia sets, fractal descriptions of bifurcations, etc), even when the syste...

Abstract. Sturmian words are infinite words that have exactly n + 1 factors of length n for every positive integer n. A Sturmian word sα,ρ is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα: x ↦ → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric proof of Yasutomi’s characterization of all pairs (α, ρ) such that sα,ρ is a fixed point of some non-trivial substitution. 1.

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 non-trivial (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.

In this paper we introduce a class of morphisms that collectively can be used to represent all Sturmian strings. 1 INTRODUCTION A string x is a concatenation of elements, called letters, drawn from a set A, called the alphabet. Throughout this paper, we confine our attention to strings x on A = fa; bg that have a well-defined leftmost position but that may extend to infinity on the right. Thus, for n 1, we represent x as an array x[1::n] with a leftmost element x[1], while for n = 0 we denote the empty string by x = ffl. We call n = jxj the length of x. A repetition is a string u k , where u is finite and nonempty and k 2 is a possibly infinite integer; that is, u concatenated with itself k \Gamma 1 times. We say also that a repetition x is periodic with period u. An infinite string x is said to be ultimately periodic iff it contains a periodic suffix. Given a string x and a letter 2A, we use the notation N (x) to denote the number of occurrences of letter in x. We say that ...