Cantor Dust Research Articles

Geometrical construction of a chaotic horseshoe-like map on Cantor dust [formula omitted]

Smale’s horseshoe map is one of the most important examples of 2D chaos in dynamical systems. Its chaotic behavior is often illustrated using a two-sided shift map. In this paper, a 3−dimensional horseshoe-like map is defined inspired by the geometrical motivation of the horseshoe map of Smale. Moreover, this construction is expressed on the Cantor dust (C×C×C) via ternary numbers and then the density of the periodic points is proved by using a different technique. To compute these periodic points more rapidly, we also give an algorithm. Finally, it is shown that this version of 3D horseshoe-like map is Devaney chaotic.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

A late-Ediacaran crown-group sponge animal.

Sponges are the most basal metazoan phylum1 and may have played important roles in modulating the redox architecture of Neoproterozoic oceans2. Although molecular clocks predict that sponges diverged in the Neoproterozoic era3,4, their fossils have not been unequivocally demonstrated before the Cambrian period5-8, possibly because Precambrian sponges were aspiculate and non-biomineralized9. Here we describe a late-Ediacaran fossil, Helicolocellus cantori gen. et sp. nov., from the Dengying Formation (around 551-539 million years ago) of South China. This fossil is reconstructed as a large, stemmed benthic organism with a goblet-shaped body more than 0.4 m in height, with a body wall consisting of at least three orders of nested grids defined by quadrate fields, resembling a Cantor dust fractal pattern. The resulting lattice is interpreted as an organic skeleton comprising orthogonally arranged cruciform elements, architecturally similar to some hexactinellid sponges, although the latter are built with biomineralized spicules. A Bayesian phylogenetic analysis resolves H. cantori as a crown-group sponge related to the Hexactinellida. H. cantori confirms that sponges diverged and existed in the Precambrian as non-biomineralizing animals with an organic skeleton. Considering that siliceous biomineralization may have evolved independently among sponge classes10-13, we question the validity of biomineralized spicules as a necessary criterion for the identification of Precambrian sponge fossils.

A combinatorial Fredholm module on self-similar sets built on $n$-cubes

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpiński carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module.

Fractal fuzzy logic using Kelly plots

Strictly defined, the concept of self-similarity or selfsimilarity applies only to mathematical fractals - which arise from the iteration of simple formulae but lead to very complex structures, Cantor Dust, Peano Curve, Koch Snowflake, whereas in natural or physical fractals - those found in nature, such as a fern leaf, an arborisation, capillaries - the concept of self-similarity applies, since their fractality is only statistical and they possess, consequently, an anisotropic scaling,(not having the same properties in all dimensions of analysis), which does not allow an amplified part of a figure to maintain exactly the characteristics of the figure as a whole, is where we find Kelly plots.

Computing the logarithmic capacity of compact sets having (infinitely) many components with the charge simulation method

We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many “small” components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in Liesen et al. (Comput. Methods Funct. Theory17, 689–713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.

A family of chaotic dynamical systems on the cantor dust C × C

In this paper, we give a family of dynamical systems on the Cantor dust (C C) by using elements of 4th Dihedral group with shift map and then express them via code representations of the points of C x C. Moreover, we prove that these systems are chaotic in the sense of Devaney and also we investigate the topological equivalence of these dynamical systems.

Fractal fuzzy logic using Kelly plots

Strictly defined, the concept of self-similarity or self-similarity applies only to mathematical fractals - which arise from the iteration of simple formulae but lead to very complex structures, Cantor Dust, Peano Curve, Koch Snowflake, whereas in natural or physical fractals - those found in nature, such as a fern leaf, an arborisation, capillaries - the concept of self-similarity applies, since their fractality is only statistical and they possess, consequently, an anisotropic scaling,(not having the same properties in all dimensions of analysis), which does not allow an amplified part of a figure to maintain exactly the characteristics of the figure as a whole, is where we find Kelly plots.

Construction of the fractal image of the “cantor dust” type using a randomized system of iterating functions

The article examines the results of the development of an algorithm for determining the number of pixels of a fractal image of the "Fractal dust" type (Cantor's set) created using resolution-limited RSIF. Mathematical formulas and schemes for calculating the number of pixels of the fractal image and the number of iterations of the RSIF fractal have been derived. This algorithm consists in finding formulas depending on the number of figures of the first iteration, similarity coefficients, and resolution. This makes it possible to determine the number of pixels for an existing fractal image with different resolutions. The calculation algorithm does not use the entry of a loop into a loop and recursive functions, and is quite optimized, which allows without spending a lot of computing power. The use of formulas in the future will make it possible to determine the number of random events (stochastic movement of a point), to ensure the quality of the reproduced image using RSIF, will make it possible to form an initial data set for neural networks, which will form the basis of object recognition.

Continuous maps that preserve Hausdorff measure

We show that the space of homeomorphisms on the unit interval [0,1], equipped with the topology of uniform convergence, contains a dense subspace of functions that preserve the Hausdorff measure of any subset of certain one-dimensional self-similar sets. We extend the results to a class of Cantor dust type self-similar sets in R2.

On arithmetic properties of Cantor sets

In this paper, we study three types of Cantor sets. For any integer m ⩾ 4, we show that every real number in [0, k] is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k, and the smallest such k is 2m. Moreover, we generalize this result to the middle-\({1 \over \alpha }\) Cantor set for \(1 < \alpha < 2 + \sqrt 5 \) and m sufficiently large. For the naturally embedded image W of the Cantor dust C × C into the complex plane ℂ, we prove that for any integer m ⩾ 3, every element in the closed unit disk in ℂ can be written as the sum of at most 2m+8m-th powers of elements in W. At last, some similar results on p-adic Cantor sets are also obtained.

Rare events for product fractal sets * *All authors were partially supported by by FCT Project PTDC/MAT-PUR/28177/2017 and by CMUP, which is financed by national funds through FCT-Fundação para a Ciência e a Tecnologia, IP, under the project with reference UIDB/00144/2020.

We analyse the existence of limiting laws of rare events for dynamically generated stochastic processes. We consider two-dimensional dynamical systems and observable functions maximised on Cantor dust sets and prove the existence of distributional limits for the partial maxima. We show how the extremal index, measuring the degree of clustering of rare events, is linked to the compatibility between the dynamics and the fractal structure of the maximal sets.

Mössbauer Probe Diagnostics of the Properties of Quasi-Liquid Water Layer on the Aluminosilicate Surface of Natural Origin

The structure and properties of the quasi-liquid water layer in the “frozen water–clay mineral” system have been studied using a probe technique developed on the basis of the 57Fe Mossbauer isotope in two forms (Fe2+ and Fe3+). The elasticity parameters (shear modulus, Poisson’s ratio, and Gruneisen parameter) of the surface ice are estimated within the fractal geometry. The “Menger sponge"–"Cantor dust” transition with an increase in the layer thickness is established.

Solutions with dense short segments from regular delays

We construct a delay functionaldU with values in (0,r) and find a positive number h<r such that the negative feedback equationx′(t)=−x(t−dU(xt,r)), with the segmentxt,r:[−r,0]→R given by xt,r(s)=x(t+s), has a solution whose short segmentsxt,h=xt,r|[−h,0] are dense in an open subset of the space C1([−h,0],R). The domain U of dU is open in C1([−r,0],R), and the delay differential equation defines a continuous semiflow of continuously differentiable solution operators on the solution manifold{ϕ∈U:ϕ′(0)=−ϕ(−dU(ϕ))}. The result implies a kind of chaotic solution behaviour which is not confined to a thin Cantor dust.

CANTOR-TYPE SETS IN HYPERBOLIC NUMBERS

The construction of the ternary Cantor set is generalized into the context of hyperbolic numbers. The partial order structure of hyperbolic numbers is revealed and the notion of hyperbolic interval is defined. This allows us to define a general framework of the fractal geometry on the hyperbolic plane. Three types of the hyperbolic analogues of the real Cantor set are identified. The complementary nature of the real Cantor dust and the real Sierpinski carpet on the hyperbolic plane are outlined. The relevance of these findings in the context of modern physics are briefly discussed.

On the fractal distribution of primes and prime-indexed primes by the binary image analysis

Abstract In this paper, the distribution of primes and prime-indexed primes (PIPs) is studied by mapping primes into a binary image which visualizes the distribution of primes. These images show that the distribution of primes (and PIPs) is similar to a Cantor dust, moreover the self-similarity with respect to the order of PIPs (already proven in Batchko (2014)) can be seen as an invariance of the binary images. The index of primes plays the same role of the scale for fractals, so that with respect to the index the distribution of prime-indexed primes is characterized by the self-similarity alike any other fractal. In particular, in order to single out the scale dependence, the PIPs fractal distribution will be evaluated by limiting to two parameters, fractal dimension ( δ ) and lacunarity ( λ ), that are usually used to measure the fractal nature. Because of the invariance of the corresponding binary plots, the fractal dimension and lacunarity of primes distribution are invariant with respect to the index of PIPs.

What is the alternative to the Alexander–Orbach relation?

The Alexander–Orbach (AO) relation dw = 2df/ds connecting the fractal dimension of a random walk’s (RW) trajectory dw or the exponent of anomalous diffusion α = 2/dw on a fractal structure with the fractal and spectral dimension of the structure itself plays a key role in discussion of dynamical properties of complex systems including living cells and single biomolecules. This relation however does not hold universally and breaks down for some structures like diffusion limited aggregates and Eden trees. We show that the alternative to the AO relation is the explicit dependence of the coefficient of the anomalous diffusion on the system’s size, i.e. the absence of its thermodynamical limit. The prerequisite for its breakdown is the dependence of the local structure of possible steps of the RW on the system’s size. The discussion is illustrated by the examples of diffusion on a Koch curve (AO-conform) and on a Cantor dust (violating AO relation).

Multifractal analysis of weighted networks by a modified sandbox algorithm.

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): “Sierpinski” WFNs and “Cantor dust” WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks — collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.

A random Cantor set as a measure of the statistical homogeneity of fractured rock masses

This study addresses the applicability of the exponent of a power-law scaling model, as an index of the statistical homogeneity of fractured rock masses. We systematically examine the capability of a random Cantor set, Dc, to quantify the combined effect of fracture density and size distribution on fracture networks. Total 129 two-dimensional fracture configurations were generated, based on different combinations of fracture size distribution and fracture density. The Dc for each generated fracture network system was calculated using the random Cantor dust method. We found that the standard deviation of trace length has no effect on the calculated Dc, and the estimated Dc increases with the total density and/or mean trace length. To explore the field applicability of this study, we investigated the statistical homogeneity of a fractured rock mass of the Upper Devonian Grosmont Formation in Alberta, Canada. The results clearly show that the methodologies given in this study are capable of determining the statistical homogeneity of fractured rock masses.

Wide band frequency selective surface for angular and polarization independent operation

ABSTRACTIn this article, the authors propose the use of Cantor dust fractal geometry design frequency selective surfaces (FSS) with wideband, angular, and polarization independent operation aiming ultrawide band (UWB) applications. The simulated results were obtained with the computer program Ansoft Designer™. A prototype of the designed FSS was built and its transmission characteristics were measured for vertical and horizontal polarization. A good agreement between simulated and measured results was obtained. The FSS was stable angularly and in terms of polarization and proved to be useful for UWB applications. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:216–219, 2015