Generalized Cantor Sets Research Articles

On the length spectrums of Riemann surfaces given by generalized Cantor sets

On the length spectrums of Riemann surfaces given by generalized Cantor sets

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.

КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ РОЗСІЮВАННЯ Е-ПОЛЯРИЗОВАНИХ ХВИЛЬ НА ЕКРАНОВАНІЙ ПРЕДКАНТОРОВІЙ СИСТЕМІ РЕФЛЕКТОРІВ

Computer simulation of electromagnetic wave scattering on a shielded periodic reflector system, with a special period structure is considered. The electrodynamic reflective structure consists of a screen and a periodic system of strips. The cross-section of the strips is a system of segments formed by the principle of constructing a generalized Cantor set at each of the periods. Іt was used a mathematical model based on the reduction of the initial boundary value problems for the Helmholtz equation to systems of singular integral equations. The choice of this approach did not require analytical calculations when changing the number of lattice elements. Using this approach, it was not necessary to make changes to the computer implementation of the algorithm. The computer experiment has proved the possibility of applying the discrete singularities method computational scheme for the considered structure. It is proved that the structure constructed according to the proposed principle has the properties of broadband and multimode for the studied values of the structure parameters. In particular, for the studied structures, the resonance occurred when the wavenumber was changed from 0 to 50, with a significant increase in amplitude occurring at harmonics with numbers up to 40. This phenomenon led to a complex structure of the lines of the absolute values of the amplitude of the scattered electric field above the grating. The presence of small areas with an increase in the amplitude of the scattered field by a factor of 5-6 compared to the amplitude the incident electromagnetic wave was observed. These properties show practical interest in the use of this type of antennas to improve electronic camouflage and tactical radio communications. In the future, it is planned to conduct computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case discussed in this paper.

A Cantor-Type Construction. Invariant Set and Measure

The basic idea we use throughout this paper is to generalize the construction of the Cantor set as the attractor of an iterated function system $\big(f_0,\ f_1\big)$, replacing the classical functions acting on $ [0,1]$ via $f_0(x) =\frac{x}{3}$ and $f_1(x)= \frac{2}{3} + \frac{x}{3}$ with the new functions acting via $f_0(x)= \theta x$ and $f_1(x) = 1 - \theta + \theta x$, where $\theta \in \Big[0,\ \frac{1}{2} \Big]$ is an arbitrary number. We add to the schema two positive numbers $p_0, \ p_1$ such that $p_0 + p_1 = 1$, obtaining the iterated function system with probabilities $\big(f_0,\; f_1 ; \; p_0,\; p_1\big)$ which generates the invariant set (attractor) and the invariant measure, according to J. Hutchinson. The structure of the invariant set (genera-lized Cantor set) being known, we focus on the detailed computation of the invariant measure. At the end of the paper, we study the dependence upon the parameter $\theta$ of the invariant set and of the invariant measure.

On the quasiconformal equivalence of dynamical Cantor sets

The complement of a Cantor set in the complex plane is itself regarded as a Riemann surface of infinite type. The problem of this paper is the quasiconformal equivalence of such Riemann surfaces. Particularly, we are interested in Riemann surfaces given by Cantor sets which are created through dynamical methods. discuss the quasiconformal equivalence for the complements of Cantor Julia sets of rational functions and generalized Cantor sets. We also consider the Teichmüller distance between generalized Cantor sets.

Fractal moiré interferometry

Fractal geometry of 2D gratings, based on a regular class of generalized Cantor sets, was proposed to derive moiré fringes with geometry induced by the fractal geometry of these gratings. The general case of a regular fractal grating based on parallel lines was considered. Some simple examples based on parallel straight lines of fractal gratings reduced to the first few steps of fractal grating geometry are briefly discussed. The formation of derivative fractal patterns of interference fringes, which at each fractal step meets the general rules for ordinary moiré patterns, is briefly discussed.

Analysis of the ground-state energy eigenvalues of fractal quantum potentials

Traditional models in solid-state physics study the motion of electrons in a periodic lattice. Recent developments in the physics of graphene allow scientists to construct two-dimensional structures with fractal geometry and conduct experiments on them. Recently, some theoretical approaches have been developed to study the optical and electrical properties of semiconductor layers with self-similar characteristics. This newly emerged direction in solid-state physics inspired us to focus on studying the quantum mechanical properties of fractal potentials. We first introduce sequences of potential wells converging towards different fractal structures. Then, we calculate the ground-state energy eigenvalues of the time-independent Schrödinger equation for these potential functions using two numerical methods, the Numerov and analytical transfer matrix method, to demonstrate the effect of the potential structure morphology and properties on the behavior of the energy eigenvalues. Ground-state energies for the generalized Cantor set, the Smith–Volterra–Cantor set, a multi-level Cantor set and the Weierstrass function will be calculated and compared. We will deal with the question of how the ground state of these potential functions changes as the fractal generator is applied, and we show that properties such as the Lebesgue measure of the Cantor potentials and the Hausdorff dimension of the Weierstrass function strongly control the convergence of energy eigenvalues.

Fractal lenses based on Cantor binary sequences for ultrasound focusing applications

In this work, we demonstrate the application of Cantor fractal lenses in acoustics. The Cantor Zone Plate (CZP), previously introduced in optics, is designed from a conventional Fresnel Zone Plate (FZP) using a binary sequence governed by the distribution of a generalized Cantor set. The CZP maintains its main focus at the same focal distance than its associated FZP, providing a softer multi-foci focusing profile which is very useful in certain ultrasound therapeutic applications. Experimental measurements are in good agreement with the theoretical model, demonstrating that CZPs are suitable for the ultrasound field.

Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps

We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit interval [0, 1] infinitely. The set of points remained in the unit interval after this removal process is over is called the Cantor set. The dimension of such a set is not an integer value. In fact, it has a ‘fractional’ dimension, making it by definition a fractal. The Cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. In this paper, we have provided three types of generalization of the Cantor set depending on the process of removal. Also, we have discussed some characteristics of the fractal dimensions of these generalized Cantor sets. Further, we have shown its application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps.

Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

We consider a class of compact positive operators $$L:X\rightarrow X$$ given by $$(Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds$$ , acting on the space X of continuous $$2\pi $$ -periodic functions x. Here $$\eta $$ is continuous with $$\eta (t)\le t$$ and $$\eta (t+2\pi )=\eta (t)+2\pi $$ for all $$t\in \mathbf{R}$$ . We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem $$\kappa x=Lx$$ exists for some $$\kappa >0$$ (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally $$\eta $$ is analytic, we study the set $${\mathcal {A}}\subseteq \mathbf{R}$$ of points t at which x is analytic; in general $${\mathcal {A}}$$ is a proper subset of $$\mathbf{R}$$ , although x is $$C^\infty $$ everywhere. Among other results, we obtain conditions under which the complement $${\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}$$ of $${\mathcal {A}}$$ is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map $$t\rightarrow \eta (t)$$ .

Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum

We construct multidimensional almost-periodic Schr\"odinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.

DNA Sequences with Forbidden Words and the Generalized Cantor Set

In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao’s frame representation and the generalized Sierpinski Set, along with their fractal graphs, are also presented in this work.

Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

In this paper, we discuss Hausdorff measure and Hausdorff dimension. We also discuss iterated function systems (IFS) of the generalized Cantor sets and higher dimensional fractals such as the square fractal, the Menger sponge and the Sierpinski tetrahedron and show the Hausdorff measures and Hausdorff dimensions of the invariant sets for IFS of these fractals.

Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities

AbstractConsider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the k-th convolution of the associated Cantor measure has local dimension at x ∊ (0,1) tending to 1 as ft: tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

Diffusion on Middle-ξ Cantor Sets.

In this paper, we study -calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the -calculus on the generalized Cantor sets known as middle- Cantor sets. We have suggested a calculus on the middle- Cantor sets for different values of with . Differential equations on the middle- Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.

Study of auroral ionosphere using percolation theory and fractal geometry

In this work, values of the fractal dimension and the connectivity index characterizing the structure of Hall conductivities on the night side of the auroral ionosphere are derived in general form. Restrictions imposed on fractal structure of the ionospheric conductivity are analyzed in terms of the percolation of the ionospheric Hall currents. It is shown that the structure of ionospheric Hall conductivities can be described as asymptotically path-connected fractal. This result is supported by analysis of typical structure observed in auroral electron precipitation which are also the main source of ionization on the night side of the ionosphere. It is demonstrated that crossing the precipitation region in the direction perpendicular to the multiple arcs system, one should observe the structure of the precipitation which looks like a generalized Cantor set.

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function ζA(s):=∫Aδd(x,A)s−Ndx, where δ>0 is fixed and d(x,A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function ζL associated with bounded fractal strings L=(ℓj)j≥1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(ζA) coincides with D:=dim‾BA, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” {Res=D}. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function |At| as t→0+, where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that |At|=tN−D(M+O(tγ)) as t→0+, with γ>0, and to a class of Minkowski nonmeasurable sets, such that |At|=tN−D(G(log⁡t−1)+O(tγ)) as t→0+, where G is a nonconstant periodic function and γ>0. In both cases, we show that ζA can be meromorphically extended (at least) to the open right half-plane {Res>D−γ} and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of ζA evaluated at s=D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line {Res=D}. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of RN, for N≥1 arbitrary. These are compact subsets of RN such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line {Res=D}.

Distance and tube zeta functions of fractals and arbitrary compact sets

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets A of the N-dimensional Euclidean space RN, for any integer N≥1. It is defined by the Lebesgue integral ζA(s)=∫Aδd(x,A)s−Ndx, for all s∈C with Res sufficiently large, and we call it the distance zeta function of A. Here, d(x,A) denotes the Euclidean distance from x to A and Aδ is the δ-neighborhood of A, where δ is a fixed positive real number. We prove that the abscissa of absolute convergence of ζA is equal to dim‾BA, the upper box (or Minkowski) dimension of A. Particular attention is payed to the principal complex dimensions of A, defined as the set of poles of ζA located on the critical line {Res=dim‾BA}, provided ζA possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, ζ˜A(s)=∫0δts−N−1|At|dt, called the tube zeta function of A. Assuming that A is Minkowski measurable, we show that, under some mild conditions, the residue of ζ˜A computed at D=dimB⁡A (the box dimension of A) is equal to the Minkowski content of A. More generally, without assuming that A is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of A. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers.

An asymptotic extension of Moran construction in metric measure spaces

In this paper, we define asymptotically generalized Cantor sets in metric measure spaces by generalizing the notion of $\lambda$-similarity maps. We define the notion of $(\lambda, c, \nu)$-similarity maps, and extend the Moran theorem about the generalized Cantor set in $\mathbf{R}^d$ to this general setting. As an example, we construct generalized Cantor sets in Riemannian manifolds by using $(\lambda, c, \nu)$-similarity maps.

On the self similarity of generalized Cantor sets

We consider the self-similar structure of the class of generalized Cantor sets $$\Gamma_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_n\beta^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<\beta<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $\Gamma_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.