Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type

Sirvent, Vı́ctor F.; Solomyak, Boris

doi:10.4153/cmb-2002-062-3

Cited by 32 publications

(42 citation statements)

References 19 publications

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“…The balanced pair algorithm was introduced by Livshits in [12] in the study of the Pisot conjecture and was also used in [21] in the same context. A variant of this algorithm was used later in [17,18] to study the intersection of Rauzy fractals associated with different substitutions having the same incidence matrix.…”

Section: Balanced Pair Algorithm and Fractals Intersectionmentioning

confidence: 99%

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Balanced pair algorithm for a class of cubic substitutions

Sellami¹

2015

Turk J Math

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Section: Balanced Pair Algorithm and Fractals Intersectionmentioning

confidence: 99%

“…For references on conditions under which the Pisot conjecture is true, we refer to, among other references, [1,2,3,4,5,6,7,11,12,15,16,21,22,23].…”

Section: Substitutions and Rauzy Fractalsmentioning

confidence: 99%

Balanced pair algorithm for a class of cubic substitutions

Sellami¹

2015

Turk J Math

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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%

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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“…There exist also effective combinatorial characterisations for pure discrete spectrum based either on graphs [307,329], or on the so-called balanced pair algorithm [246,311], or else conditions inspired by the strong coincidence condition [46,55,54,190]. More generally, for more on the spectral study of substitutive dynamical systems, see [173,145].…”

Section: The Pisot Conjecturementioning

confidence: 99%