Abstract not found

We consider discrete one-dimensional Schrodinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ff-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

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...

this paper is to present a new proof, with some improvements, of a theorem by Mignosi [21] cited below. Let x be an infinite word, and let F (x) be the sets of its factors (subwords). For w 2 F (x), the index of w in x is the greatest integer d such that w

Communicated byM. Waldschmidt

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 describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence we obtain a complete characterization of those sequences of block complexity 2n + 1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.

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.

A striking conjecture of Fraenkel asserts that the only decomposition of Z into m 3 sequences fb i n+ i cg n2Z with i and i real, i > 1 and i 's distinct for i = 1; :::; m is given by f 1 ; :::; m g = f 2 m 1 2 k : 0 k < mg: A weaker conjecture states that then i = j is an integer for some i 6= j. The weaker statement has been proved by Simpson if some i is at most 2. Fraenkel's conjecture was proved by Simpson if some i is at most 3=2 and by Morikawa if m = 3 and, under some condition, if m = 4. An unconditional proof for m = 4 was given by Altman, Gaujal and Hordijk and for m = 5; 6 by the author. Proofs for the more general case of balanced sequences have been given for m = 3 by the author and for m = 4 by Altman, Gaujal and Hordijk. In the present paper we extend the other results mentioned above to balanced sequences. We start with a survey of the related literature and give self-contained proofs. Keywords: complementary sequences, eventual coveri...

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patch-counting function NX(T) of radius T being finite for all T. A Delone set X of finite type is repetitive if there is a function MX(T) such that every closed ball of radius MX(T)+T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R n-action. There is a lower bound for MX(T) in terms of NX(T), namely MX(T) ≥ c(NX(T)) 1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T) purely in terms of NX(T). The complexity of a repetitive Delone set of finite type is measured by the growth rate of its repetitivity function MX(T). For example, the function MX(T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T) = O(T) as T → ∞ and is densely repetitive if MX(T) = O(NX(T)) 1/n as T → ∞. We show that linearly repetitive sets