Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux, Pierre; Berthé, Valérie; Ito, Shunji

doi:10.5802/aif.1889

Cited by 48 publications

(64 citation statements)

References 19 publications

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“…non-integer base, and yield certain generalizations of Galois' theorem [27,30,57]. In discrete geometry, there are numerous relations between generalized Rauzy fractals and discrete planes as studied for instance in [13]. The shape of pieces generating a discrete plane is tightly related to the shape of Rauzy fractals.…”

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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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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“…In the irreducible case, this theorem is proved in [4] by associating the stepped-surface with a certain dynamical system on the real line. We do not know whether their method still works for the reducible case.…”

Section: Theorem 14 -The Stepped-surface S Of a Pisot Substitution Imentioning

confidence: 99%

Atomic surfaces, tilings and coincidences II. Reducible case

Ei¹,

Ito²,

Rao³

2006

Ann. inst. Fourier

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“…-In discrete geometry, there are numerous relations between generalized Rauzy fractals and discrete planes as studied for instance in [22]. The shape of pieces generating a discrete plane is tightly related to the shape of Rauzy fractals.…”

Section: The Geometry Of Substitutions: Rauzy Fractalsmentioning

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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Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Cited by 48 publications

References 19 publications

Decidability Problems for Self-induced Systems Generated by a Substitution

Decidability Problems for Self-induced Systems Generated by a Substitution

Atomic surfaces, tilings and coincidences II. Reducible case

Topological properties of Rauzy fractals

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