Results 1  10
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58
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary 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 ..."
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Cited by 68 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary 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 CobhamSemenov, the original proof being published in Russian. 1
Cubic Pisot units with finite beta expansions
 In Algebraic number theory and Diophantine analysis
, 1998
"... 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. ..."
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Cited by 35 (4 self)
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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
A generalization of Sturmian sequences; combinatorial structure and transcendence
 Acta Arith
"... In this paper we study dynamical properties of a class of uniformly recurrent sequences on a kletter 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 ..."
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Cited by 33 (5 self)
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In this paper we study dynamical properties of a class of uniformly recurrent sequences on a kletter 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 ArnouxRauzy 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 Sadic description of the characteristic sequence similar to that given by Arnoux and Rauzy for k = 2, 3. ArnouxRauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every ArnouxRauzy 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 bdigit expansion is an ArnouxRauzy sequence, is transcendental. This yields a class of transcendental numbers of arbitrarily large linear complexity. I
Structure of three interval exchange transformations II: A combinatorial description of . . .
, 2002
"... 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 const ..."
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Cited by 28 (5 self)
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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 twodimensional 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 threeinterval exchange transformations, thereby answering an old question of Rauzy.
On The Boundary Of Self Affine Tilings Generated By Pisot Numbers
 J. Math. Soc. Japan
"... 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 som ..."
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Cited by 25 (6 self)
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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.
Numeration systems and Markov partitions from self similar tilings
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the int ..."
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Cited by 24 (0 self)
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Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms. Fractals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of R 2 with numeration systems for R 2 in complex bases [16], [8]. We will see that fractal self similar tilings of R n provide natural building blocks for numeration systems of R n. These numeration systems generalize the 1dimensional cases in [14],[10],[11] as well as the 2dimensional cases mentioned above.
Complexity of sequences defined by billiards in the cube
 Bull. Soc. Math. France
, 1994
"... 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 ..."
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Cited by 23 (0 self)
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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 sousmots 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.
Selfaffine tiling via substitution dynamical systems and Rauzy fractals
 Pacific J. Math
"... 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 selfaffine multitiles and selfaffine tilings. This provides an efficient and unconventional way for constructing aperiodic selfaffine tilings. Our result ..."
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Cited by 15 (5 self)
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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 selfaffine multitiles and selfaffine tilings. This provides an efficient and unconventional way for constructing aperiodic selfaffine tilings. Our result also leads to a proof that a Rauzy fractal R associated with a primitive and unimodular Pisot substitution has nonempty interior.
Geometric Realizations Of Substitutions
 BULL. SOC. MATH. FRANCE
, 1998
"... 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 ..."
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Cited by 15 (4 self)
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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.