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 consider Sturmian sequences and provide an explicit formula for the index of such a sequence in terms of the continued fraction expansion coecients of its slope.

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.

A word u is called 1-balanced if for any two factors v and w of u of equal length, we have 1 jvj i jwj i 1 for each letter i, where jvj i denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multi-dimensional words. We rst characterize all 1-balanced words on Z n . In particular we prove they are fully periodic for n > 1. We then give a quantitative measure of balancedness for some words on Z 2 with irrational density, including twodimensional Sturmian words. 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.

We consider Sturmian sequences and explicitly determine all the integer powers occurring in them. Our approach is purely combinatorial and is based on canonical decompositions of Sturmian sequences and properties of their building blocks.

One of the numerous characterizations of Sturmian words is based on the notion of balance. An infinite word x on the f0; 1g alphabet is balanced if, given two factors of x, w and w 0 , having the same length, the difference between the number of 0 0 s in w (denoted by jwj0) and the number of 0 0 s in w 0 is at most 1, i.e. jjwj0 \Gamma jw 0 j 0 j 1. It is well known that an aperiodic word is Sturmian if and only if it is balanced. In this paper, the balance notion is generalized by considering the number of occurrences of a word u in w (denoted by jwju) and w 0 . The following is obtained Theorem Let x be a Sturmian word. Let u, w and w 0 be three factors of x. Then, jwj = jw 0 j =) fi fi jwju \Gamma jw 0 j u fi fi juj: Another balance property, called equilibrium, is also given. This notion permits us to give a new characterization of Sturmian words. The main techniques used in the proofs are word graphs and return words. 1 Introduction Sturmi...

We study powers of prefixes of Sturmian words. 1.