Figure 1 - uploaded by Klaus Scheicher
Content may be subject to copyright.
Source publication
The Twin Dragon and Rauzy fractals are intersected with the real axis. In the Twin Dragon case, unexpectedly from its fractal nature, the intersection is an interval characterized by a finite automaton. For the case of the Rauzy fractal, it is proved that the intersection has infinitely many components.
Context in source publication
Similar publications
Myth is a flexible form of communication by oral means, which has developed traditionally by word of mouth through stories, which can be an entertainment attraction for tourists. One of them is Komodo Island, Labuan Bajo which is a tourist attraction is Komodo which is included in the 7 wonders of the world (Seven wonders) and a growing myth that K...
Citations
... In number theory, diophantine properties are induced by properties of a distance function to a specific broken line [53] related to the Rauzy fractal and the size of the largest ball contained in it. 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]. ...
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 paths in A × A are paths in A and A , so it is said that the paths in the product automaton are the common paths in each of the factors. The product automaton was used in [20, 22, 23], in the context of substitution dynamical systems and Rauzy fractals, and in [3] in the study of the twin-dragon fractal. The automaton A n × A n , the product of A n with its dual, is shown inFigure 2, for n > 1. ...
We consider a family of dynamically defined subsets of Rauzy fractals in the plane. These sets were introduced in the context of the study of symmetries of Rauzy fractals. We prove that their Hausdorff dimensions form an ultimately increasing sequence of numbers converging to 2. These results answer a question stated by the third author in 2012.
... √ 5 2 b implies γ(β) = 0? (C) What is the structure of the prefixes of β-adic expansions of integers for a general quadratic β? (D) What about the cubic Pisot case? Akiyama and Scheicher [AS05] showed how to compute the value γ(β) for β ≈ 1.325 the minimal Pisot number (or Plastic number), root of β 3 = β + 1. Loridant et al. [LMST13] gave the contact graph of the β-tiles for cubic units, which could be used to determine γ(β) for the units, in a similar way to what Akiyama and Scheicher did. The consideration of the β-adic spaces could then allow the results to be expanded to non-units as well. ...
We study rational numbers with purely periodic R\'enyi $\beta$-expansions.
For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give
a necessary and sufficient condition for $\gamma(\beta)=1$, i.e., that all
rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic
$\beta$-expansion. A simple algorithm for determining the value of
$\gamma(\beta)$ for all quadratic Pisot numbers $\beta$ is described.
... What should be taken into consideration is the growth towards infinity of the ratio, and clearly the fractal dimension will show up. Some ideas about these questions can be found, e.g., in [58, 35, 1, 44, 40, 41], in particular in relation with the entropy of a curve, as discussed in several papers of Mendès France. We will conclude this paper with that notion of entropy for a plane curve. ...
In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy–)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent than other criteria for measuring its variational complexity. We also compute the inconstancy of classical binary sequences including some automatic sequences and Sturmian sequences.
... Proof. We use an argument similar to the one used in [1], in which Akiyama and Scheicher determined the intersections of the twin-dragon and the axes. By Lemma 2, any integer x can be uniquely represented as ...
A tiling in a finite abelian group H is a pair (T,L) of subsets of H such that any h∈H can be uniquely represented as t+l where t∈T and l∈L. This paper studies a finite analogue of self-affine tilings in Euclidean spaces and applies it to a problem of broadcasting on circuit switched networks. We extend the tiling argument of Peters and Syska [Joseph G. Peters, Michel Syska, Circuit switched broadcasting in torus networks, IEEE Trans. Parallel Distrib. Syst., 7 (1996) 246–255] to 3-dimensional torus networks.
... Another example providing a relation between fractals and numeration is given by the local spiral shape of the boundary of the Hokkaido tile: by constructing a realization of the natural extension for the so-called beta-transformation, one proves that unexpected non-uniformity phenomena appear in the beta-numeration associated with the smallest Pisot number. Indeed, the smallest positive real number that can be approximated by rational numbers with non-purely periodic beta-expansion is an irrational number slightly smaller than 2/3 (see [1, 10]). All the fractals mentioned so far are examples of the new class of tiles associated with shift radix systems which forms the main object of the present paper.Figure 1. Examples of SRS tiles: The upper left tile is the so-called twin dragon fractal (cf. ...
Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.
In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.
We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).
... Indeed, he obtained that the smallest Pisot number η, which is the real root of the polynomial x 3 − x − 1, satisfies 0 < γ(η) < 1. More precisely, it was proved in [8] that γ(η) is abnormally close to the rational number 2/3 since one has γ(η) = 0.666 666 666 086 · · · . ...
We study real numbers β with the curious property that the β-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have
to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize
algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by
Schmidt, Hama and Imahashi. Let γ(β) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic β-expansion. We prove that γ(β) is irrational for a class of cubic Pisot units that contains the smallest Pisot number η. This result is motivated by the observation of Akiyama and Scheicher that γ(η) = 0.666 666 666 086 … is surprisingly close to 2/3.
... Pour choisir une bonne partition pour les automorphismes et les additions sur un tore, nous venons de voir qu'il est naturel de considérer une structure compacte autosimilaire appelée fractal de Rauzy, au moins dans le cas où l'application considérée est décrite par un nombre algébrique unitaire dont tous les conjugués sont de module inférieur à 1. Vérifier que cette structure compacte est adaptée pour représenter l'application revient à vérifier une propriété de pavage, question pour laquelle j'ai proposé d'utiliser un graphe de frontière qui décrit l'intersection du fractal avec ses voisins dans un recouvrement localement fini. Contribution sur la topologie des fractals Les propriétés topologiques des fractals s'interprètent souvent en propriétés dans les différents domaines abordés, comme nous le détaillerons dans la section 3.1 : en théorie des nombres, il s'agit de caractériser des développements finis ou purement périodiques [AS05,IR05]. En dynamique, la connexité de la base de la partition est un élément important pour décider si cette partition de Markov est génératrice [Adl98]. ...
... -Dans un travail en cours avec B. Adamczewski, W. Steiner et C. Frougny (A., F., S.& S., en cours), nous avons utilisé la description de la frontière par le fractal de Rauzy pour montrer que dans le cas cubique unitaire γ(β) est un nombre irrationnel. En particulier, nous prouvons ainsi que la quantité 0.66666666608644067488 estimée dans [AS05] est bien l'approximation d'un nombre irrationnel. Il faut bien noter que les preuves d'irrationalité faisant appel à des considérations topologiques sont extrêmement rares. ...
... En théorie des nombres, des propriétés diophantiennes (recherche des meilleures approximations simultanées de vecteurs pour certains nombres) se déduisent de la largeur de la plus grande boule centrée en zéro et incluse dans le fractal. Par ailleurs, le fait que 0 est point intérieur du fractal signifie que les écritures de nombres dans une numération en base β sont toutes finies pour les éléments dans l'anneau R[1/β][AS05]. De plus, le fractal de Rauzy permet de caractériser les rationnels dont le développement est purement périodique[IR05]. Nous reviendrons longuement sur ces deux derniers points dans la section 3.2, en discutant des travaux (publié ou en cours de rédaction) en collaboration avec V. Berthé, S. Akiyama, G.Barat, B. Adamczewski, C. Frougny et W. Steiner ((Monas. ...
Applications focus on several mathematical fields and molecular biology. Our purpose is to illustrate how one can couple algorithmic approaches with discretization methods of dynamical system in order to exploit (even partial) knowledge on the system.
The first type of system we adress consist in toral autormorphisms and translations. To obtain informations on their applications needs to abstract their dynamical properties. Inspired by the one-dimensional case (beta-numeration and sturmian sequences); the main question is to exhibit a toral fundamental domain and a partition of this domain in which the dynamics of the systems is coded by simple symbolic systems. We used a decidable approach to describe the fractal boundary of the domain in the Pisot case and its tiling properties and we derived criteria to decide whether it generates a good coding. From their topological properties we deduced applications in number theory (characterization of rational numbers with a finite or purely periodic expansion in a non-integer basis) and discrete geometry (condition for the iterative generation of discrete planes).
The second class of question we adress in the framework of large-scale dynamical systems in molecular biology is how to produce information on a system that is only partially known and observed. We introduced a formalism to interpret observations in molecular biology, with a dynamical background. The main purpose was to guide model correction and, in the future, experiment design. Qualitative aspects are replaced by properties of shift equilibria,
so that analyzing data is reduced to writting and then solving constraints on discrete sets.
... This seems reasonable at least in the unit case, but requires a precise study of the topology of the central tile. Examples of computations of intersections between line and fractals are obtained in [AS05], by numeric approximations. As an example, an intersection is prove to be approximated by 0.66666666608644067488. ...
This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar representation of numbers having no fractional part in their β-expansion, called central tile: for elements x of the ring ℤ[1/β], so-called x-tiles are introduced, so that the central tile is a finite union of x-tiles up to translation. These x-tiles provide a covering (and even in some cases a tiling) of the space we are working in. This space, called complete representation space, is based on Archimedean as well as on the non-Archimedean completions of the number field ℚ(β)corresponding to the prime divisors of the norm of β. This representation space has numerous potential implications. We focus here on the gamma function γ(β) defined as the supremum of the set of elements v in [0, 1] such that every positive rational number p/q, with p/q ≤ v and q coprime with the norm of β, has a purely periodic β-expansion. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called "boundary graph". The paper ends with explicit quadratic examples, showing that the general behaviour of γ(β) is slightly more complicated than in the unit case.
... This is just a consequence of the fact that the origin is an exclusive inner point of T λ under (F) condition. For a concrete case, we can show a strange phenomenon ([10]): (0, 1), that all rationals in [ The later statement reflects the fractal structure of the boundary of T λ and perhaps it is not so easy to obtain this conclusion in a purely algebraic manner. This type of tight connection between fractal geometry and number theory is one of the aim of our research. ...
Number systems in Pisot number base are discussed in relation to arith-metic construction of quasi-crystal model. One of the most important ideas is to introduce a 'dual tiling' of this system. This provides us a geometric way to under-stand the 'algebraic structure' of the above model as well as dynamical understand-ing of arithmetics algorithms. 1. Beta expansion and Pisot number system For this section, the reader finds a nice survey by Frougny [41]. However we give a brief review and concise proofs of fundamental results to make this note more self-contained. Let us fix β > 1 and A = [0, β) ∩ Z. Denote by A * the set of finite words over A and by A N the set of right infinite words over A.
