Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts

Baker, Veronica; Barge, Marcy; Kwapisz, Jarosław

doi:10.5802/aif.2238

Cited by 31 publications

(52 citation statements)

References 25 publications

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“…-Checking whether the Rauzy fractal generates self-similar tiling of the plane is decidable [7,8,17,43,58,69,70,85]. -The box-counting dimension of the fractal boundary of the Rauzy fractal and its subtiles is computable.…”

Section: Decidability Of Rauzy Fractals Propertiesmentioning

confidence: 99%

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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: Decidability Of Rauzy Fractals Propertiesmentioning

confidence: 99%

“…Importantly, in this setting, connectivity properties of Rauzy fractals are crucial to establish generator properties of the Markov partition [3]. In tiling theory, Rauzy fractals are used to represent the tiling flow and to prove that substitutive systems are expanding foliations of the space tiling [17].…”

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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“…Substantial literature is devoted to Conjecture 4.9 which is reviewed in [272], Chap.7. See also [55,54,46,65,190] for recent results.…”

Section: The Pisot Conjecturementioning

confidence: 99%

Dynamical directions in numeration

Barat¹,

Berthé²,

Liardet³

et al. 2006

Ann. inst. Fourier

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“…Following this paper, several works are devoted to this topic, namely, Berthé and Siegel [11], Baker, Barge and Kwapisz [6] and the present paper. These works are carried out more or less independently, with emphasis on different aspects.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…A numeration system due to Dumont and Thomas [13], plays an central role in [11], and the discussion is number theoretical. The paper [6] mainly concerns the spectral aspect and the super-coincidence condition.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%