Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

Akiyama, Shigeki; Barat, Guy; Berthé, Valérie; Siegel, Anne

doi:10.1007/s00605-008-0009-7

Cited by 24 publications

(32 citation statements)

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“…Furthermore, if x is a quadratic number, and x stands for its algebraic conjugate, then the continued fraction expansion of x is purely periodic if and only if x is an irrational quadratic number and x < −1: this is Galois' theorem. Note that points having periodic orbits are dense in [0,1]. This is also the case of points having finite orbits: these are exactly the rational numbers in [0, 1].…”

Section: Periodic Orbits and Dynamicsmentioning

confidence: 96%

“…The analogue of Lagrange theorem has been proved in [14,89]: if β is a Pisot number, then x ∈ [0, 1] has an eventually periodic expansion if and only if x ∈ Q(β) ∩ [0, 1]. Moreover, an analogue of Galois' theorem is given in [1] (see also the references therein): let β be a Pisot number; a real number x ∈ Q(β) ∩ [0, 1) has a purely periodic β-expansion if and only if x and its conjugates belong to an explicit subset in a finite product of Euclidean and p-adic spaces that depends on β; this set (called generalized Rauzy fractal) is a graph-directed self-affine compact subset of non-zero measure; the primes p that occur are prime divisors of the norm of β. The proof is here again based on an explicit realization of the natural extension of the β-transformation T β .…”

Section: Periodic Orbits and Dynamicsmentioning

confidence: 99%

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Numeration and discrete dynamical systems

Berthé

2011

Computing

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This survey aims at giving both a dynamical and computer arithmetic-oriented presentation of several classical numeration systems, by focusing on the discrete dynamical systems that underly them: this provides simple algorithmic generation processes, information on the statistics of digits, on the mean behavior, and also on periodic expansions (whose study is motivated, among other things, by finite machine simulations). We consider numeration systems in a broad sense, that is, representation systems of numbers that also include continued fraction expansions. These numeration systems might be positional or not, provide unique expansions or be redundant. Special attention will be payed to β-numeration (one expands a positive real number with respect to the base β > 1), to continued fractions, and to their Lyapounov exponents. In particular, we will compare both representation systems with respect to the number of significant digits required to go from one type of expansion to the other one, through the discussion of extensions of Lochs' theorem.

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Section: Periodic Orbits and Dynamicsmentioning

confidence: 96%

Section: Periodic Orbits and Dynamicsmentioning

confidence: 99%

Numeration and discrete dynamical systems

Berthé

2011

Computing

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“…Finiteness properties of digit representations in numeration systems with non-integer base are related to the fact that 0 is an inner point of the Rauzy fractal [9]. More generally, the identification of those real numbers who has a periodic expansion in non-integer basis is strongly related to the study of the intersection of the fractal boundary with appropriate lines [2,6]. Rauzy fractals also allow one to characterize purely periodic orbits of representations in numeration systems w.r.t.…”

Section: The Geometry Of One-dimensional Substitutionsmentioning

confidence: 99%

Decidability Problems for Self-induced Systems Generated by a Substitution

Jolivet

Siegel

2015

Lecture Notes in Computer Science

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Abstract. In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.The following survey is mainly inspired by three papers from the authors and their collaborators [32,60,84].

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“…This geometrical characterization builds a bridge between number theory and topology. As an example, based on our geometrical characterization, the geometric property (F) implies that all rational number sufficiently near to zero have a purely periodic beta-expansion in the primitive unit Pisot case [10,82]. In other words, the topology of the central tiles (in particular, the question whether 0 is an inner point or not) relates to unexpected properties of beta-numeration (note that the behavior of rational numbers with respect to purely periodic beta-expansion is far from random).…”

Section: Number Theorymentioning

confidence: 99%

“…To go further, we know that when β is still Pisot but not a unit, a suitable natural extension is not built from the central tile itself but it requires additional p-adic components (see [10,41]). In spite of that, the construction of the natural extension remains very similar to the unit case and boundary graphs can be defined as well in this situation.…”

Section: Number Theorymentioning

confidence: 99%

Topological properties of Rauzy fractals

Siegel¹,

Thuswaldner²

2009

Mémoires Soc. Math. France

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Abstract. -Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions.To be more precise, let σ be a substitution over the finite alphabet A. We assume that the incidence matrix of σ is primitive and its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to σ a set T which is now called central tile or Rauzy fractal of σ. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n = #A subtiles T (1), . . . , T (n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary.Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and geometry of the underlying central tile.After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of T and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether T and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a disk and derive properties of their fundamental group.The basic tools for our criteria are several classes of graphs built from the description of the tiles T (i) (1 ≤ i ≤ n) as graph directed iterated function systems and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries ∂T as well as ∂T (i) (1 ≤ i ≤ n) and all points where two, three or four different tiles of the induced tilings meet.When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for "everyday's life" when dealing with topological and geometric properties of substitutions.Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

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Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

Cited by 24 publications

References 28 publications

Numeration and discrete dynamical systems

Numeration and discrete dynamical systems

Decidability Problems for Self-induced Systems Generated by a Substitution

Topological properties of Rauzy fractals

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