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157
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
The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be som ..."
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Cited by 59 (9 self)
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We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
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
BetaIntegers as Natural Counting Systems for Quasicrystals
, 1998
"... . Recently, discrete sets of numbers, the fiintegers ZZ fi , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational fi ? 1 in which the fiintegers are all real numbers with no fractional part. These fiintegers app ..."
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Cited by 34 (4 self)
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. Recently, discrete sets of numbers, the fiintegers ZZ fi , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational fi ? 1 in which the fiintegers are all real numbers with no fractional part. These fiintegers appear as being quite appropriate to describing some quasilattices relevant to quasicrystallography when precisely fi is equal to 1+ p 5 2 (golden mean ), to 1+ p 2, or to 2+ p 3, i.e. when fi is one of the selfsimilarity ratios observed in quasicrystalline structures. As a matter of fact, fiintegers are natural candidates for coordinating quasicrystalline nodes, and also the Bragg peaks beyond a given intensity in corresponding dioeraction patterns: they could play the same role as ordinary integers do in crystallography. In this paper, we prove interesting algebraic properties of the sets ZZ fi when fi is a iquadratic unit PV numberj, a class of algebraic integers which includes t...
Ergodic properties of Erdős measure, the entropy of the goldenshift and related problems
 MATH
, 1998
"... We define a twosided analog of Erdös measure on the space of twosided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T² which is the induced ..."
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Cited by 32 (17 self)
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We define a twosided analog of Erdös measure on the space of twosided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on T² which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdős measure on the interval, its entropy in the sense of GarsiaAlexanderZagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
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.
Measure and Dimension for some Fractal Families
 Proc. Cambridge Phil. Soc
, 1998
"... We study selfsimilar sets with overlaps, on the line and in the plane. It is shown that there exist selfsimilar sets that have nonintegral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parame ..."
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Cited by 24 (8 self)
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We study selfsimilar sets with overlaps, on the line and in the plane. It is shown that there exist selfsimilar sets that have nonintegral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parameter value. We also explore conditions for the validity of Falconer's formula for the Hausdorff dimension of selfaffine sets, and study the dimension of some fractal graphs. Introduction In this paper we study some families of fractals depending on parameters. It turns out that while individual members of the family are often mysterious, some information can be obtained about the "typical" member. The reader is referred to [12, 23] for the background in dimension theory, and to [37] for a survey of recent results related to this paper. In the first two sections we consider homogeneous selfsimilar sets whose construction involves overlaps. In particular, let C = C [ (C + 1) [ (C + 3) = ...
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.
Arithmetic Construction of Sofic Partitions of Hyperbolic Toral Automorphisms
 Th. & Dynam. Sys
, 1998
"... For each irreducible hyperbolic automorphism A of the ntorus we construct a sofic system (\Sigma; oe) and an isomorphism of (\Sigma; oe) with a finite cover of (T n ; A). This construction is natural in the sense that it depends only on the characteristic polynomial of A, and furthermore it has a ..."
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Cited by 22 (3 self)
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For each irreducible hyperbolic automorphism A of the ntorus we construct a sofic system (\Sigma; oe) and an isomorphism of (\Sigma; oe) with a finite cover of (T n ; A). This construction is natural in the sense that it depends only on the characteristic polynomial of A, and furthermore it has an arithmetic interpretation. 1 Introduction Starting from [1, 3, 9], many papers have been devoted to Markov partitions for hyperbolic automorphisms of the torus, as well as Markov partitions for arbitrary hyperbolic systems. If we want to give a symbolic realization of such a toral automorphism A, the following question naturally arises: is it possible to interpret other structures which exist on the torus (foliations, homoclinic points, algebraic and arithmetic structures) in terms of the symbolic dynamics? The constructions of [3, 9] are very general (they work for arbitrary Anosov systems) and do not respond to this question. CNRS UMR 128, Ecole Normale Sup'erieure de Lyon, 46, all'e...
Thuswaldner, Generalized radix representations and dynamical systems
 II, Acta Arith
"... Abstract. For r = (r1,..., rd) ∈ R d the map τr: Z d → Z d given by τr(a1,..., ad) = (a2,..., ad, −⌊r1a1 + · · · + rdad⌋) is called a shift radix system if for each a ∈ Zd there exists an integer k> 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix system ..."
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Cited by 21 (11 self)
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Abstract. For r = (r1,..., rd) ∈ R d the map τr: Z d → Z d given by τr(a1,..., ad) = (a2,..., ad, −⌊r1a1 + · · · + rdad⌋) is called a shift radix system if for each a ∈ Zd there exists an integer k> 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain wellknown notions of number systems like βexpansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials d�i piX =0 i ∈ Z[X] of degree d with a large but fixed constant term p0 approximate the set of (d − 1)dimensional shift radix systems. The proofs make extensive use of the following tools: Firstly, vectors r ∈ Rd which define shift radix systems are strongly connected to monic real polynomials all of whose roots lie inside the unit circle. Secondly, geometric considerations which were established in Part I of this series of papers are exploited. The main results establish two conjectures mentioned in Part II of this series of papers. 1.