We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian. 1

Abstract. Cubic Pisot units with finite beta expansion property are classified (Theorem 3). The results of [6] and [3] are well combined to complete its proof. Further, it is noted that the above finiteness property is equivalent to an important problem of fractal tiling generated by Pisot numbers. 1991 Mathematics Subject Classification: 11K26,11A63,11Q15,28A80

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

Denition and fundamentals of tilings generated by Pisot numbers are shown by arithmetic consideration. Results include the case that a Pisot number does not have a nitely expansible property, i.e. a soc Pisot case. Especially we show that the boundary of each tile has Lebesgue measure zero under some weak condition. 1.

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.

RÉSUMÉ. — Nous démontrons une conjecture de Gérard Rauzy relative à la structure des trajectoires de billard dans un cube. A chaque trajectoire on associe la suite à valeurs dans {1, 2, 3} obtenue en codant par un 1 (resp. 2, 3) chaquerebond sur une paroi frontale (resp. latérale, horizontale). Nous montrons que si la direction initiale est totalement irrationnelle, le nombre de sous-mots distincts apparaissant dans cette suite est exactement n2 + n +1. ABSTRACT. — We prove a conjecture of Gérard Rauzy related to the structure of billiard trajectories in the cube: let us associate to any such trajectory the sequence with values in {1, 2, 3} given by coding 1 (resp. 2, 3) any time the particle rebounds on a frontal (resp. lateral, horizontal) side of the cube. We show that, if the direction is totally irrational, the number of distinct finite words of length n appearing in this sequence is exactly n2 + n +1. 1. Statement of the result We consider billiard problems with elastic reflexion on the boundary; the simplest of these problems is the billiard in the square. It is natural to code an orbit for this billiard problem with initial direction (α, β) by the sequence of the sides it meets, coding 1 for vertical sides and 2 for (*) Texte reçu le 14 janvier 1992.

Substitution dynamical systems are abstract objects and it is therefore natural to look for ways of representing them geometrically. In this article we give geometric realizations for a large class of substitutions. We only require that the substitution be primitive and that the incidence matrix have a nonzero eigenvalue of modulus less than one.

In this paper we show that a class of sets known as the Rauzy fractals, which are constructed via substitution dynamical systems, give rise to self-affine multi-tiles and self-affine tilings. This provides an efficient and unconventional way for constructing aperiodic self-affine tilings. Our result also leads to a proof that a Rauzy fractal R associated with a primitive and unimodular Pisot substitution has nonempty interior.

Abstract not found

ABSTRACT. V. Berthé showed that the frequencies of factors in a Sturmian word of slope α, as well as the number of factors with a given frequency, can be expressed in terms of the continued fraction expansion of α. In this paper we describe a multi-dimensional continued fraction process associated with a class of sequences of (block) complexity kn+1 originally introduced by P. Arnoux and G. Rauzy. This vectorial division algorithm yields simultaneous rational approximations of the frequencies of the letters. We extend Berthé’s result to factors of Arnoux-Rauzy sequences by expressing both the frequencies and the number of factors with a given frequency, in terms of the ‘convergents ’ obtained from the generalized continued fraction expansion of the frequencies of the letters. 1.