Mandelbrot Set Research Articles

A New Chaotic Image Steganography Technique Based on Huffman Compression of Turkish Texts and Fractal Encryption With Post-Quantum Security

As the digital age is becoming an unavoidable part of our daily life people are becoming more concern about their privacy and security of their digital communications. Steganography and cryptography are the two most frequently used merits of digital life to fulfill these needs. In this study, a new spatial domain chaotic steganography scheme is presented utilizing also new fractal stream encryption algorithm to hide Huffman encoded and compressed Turkish texts. The study composes of four main phases. Firstly, a sample of Turkish newspaper columnist corpus is gathered to obtain not only the frequencies of letters including special Turkish characters but also punctuations, spaces, newlines, quotations, etc. As a result of the first phase, a static Huffman encoding dictionary is obtained for 102 encountered characters. Secondly, super Mandelbrot sets are used with the Logistic map to obtain one-time-pad stream keys to encrypt compressed texts. Thirdly, the LSB (Least Significant Bit) plain of the cover image is analyzed morphologically to find low entropy pixel locations so that in steganography phase spotted locations can be avoided and scheme can become more resistant against stego analysis. Lastly, the chaotic steganography phase is the final phase in which data hiding pixel is selected according to logistic map chaotic function then LSB of the selected pixel is updated. Many tests are carried out on many images to show the image quality namely PSNR, SNR, SSMI, UIQI, Entropy, MSE, and histogram. Results indicate that proposed schemes are successful for encryption and offer robust steganography.

Simple proofs for the derivative estimates of the holomorphic motion near two boundary points of the Mandelbrot set

For the complex quadratic family qc:z↦z2+c, it is known that every point in the Julia set J(qc) moves holomorphically on c except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points 1/4 and −2: for each z=z(c) in the Julia set, the derivative dz(c)/dc is uniformly O(1/1/4−c) when real c↗1/4; and is uniformly O(1/−2−c) when real c↗−2. These estimates of the derivative imply Hausdorff convergence of the Julia set J(qc) when c approaches these boundary points. In particular, the Hausdorff distance between J(qc) with 0≤c<1/4 and J(q1/4) is exactly 1/4−c.

Perfect subspaces of quadratic laminations

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady (1984) and, separately, by Thurston (1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by ipinching the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be irenormalized to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.

Generalized Rings Around the McMullen Domain

We consider the family of rational maps given by $$F_\lambda (z)=z^n+\lambda /z^d$$ where $$n, d \in \mathbb {N}$$ with $$1/n+1/d<1,$$ the variable $$z\in {\widehat{{\mathbb {C}}}}$$ and the parameter $$\lambda \in {\mathbb {C}}$$ . It is known that when $$n=d \ge 3$$ there are infinitely many rings $${\mathcal {S}}^k$$ with $$k\in \mathbb {N}$$ , around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter $$\lambda $$ -plane where the Julia sets of $$F_\lambda $$ are Cantor sets of simple closed curves. The rings $${\mathcal {S}}^k$$ converge to the boundary of the McMullen domain as $$k \rightarrow \infty $$ and contain parameter values that lie at the center of Sierpinski holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of $$F_\lambda $$ are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when $$1/n+1/d<1$$ where n is not necessarily equal to d. The number of Sierpinski holes and superstable parameters on $${\mathcal {S}}^1$$ is $$\tau _1^{n,d} = n-1,$$ and on $${\mathcal {S}}^k$$ for $$k> 1$$ is given by $$\tau _k^{n,d} = dn^{k-2}(n-1)-n^{k-1} + 1$$ .

The local Hölder exponent for the entropy of real unimodal maps

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ (equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Holder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Holder exponent equals exactly, up to a factor log 2, the value of the function at that point. This confirms a conjecture of Isola and Politi (1990), and extends a similar result for the dimension of invariant subsets of the circle.

FRACTALS FROM NONLINEAR IFSs OF THE COMPLEX MAPPING FAMILY f(z) = zn + c

To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family [Formula: see text] ([Formula: see text]). A set of [Formula: see text]-values is chosen from the period-1 bulb of the Mandelbrot set, so that each mapping has an attracting fixed point in the dynamic plane. Computer experiments show that a set of arbitrarily chosen [Formula: see text]-values may not be able to generate a fractal. We prove a sufficient condition that if the [Formula: see text]-values are chosen from a specific region related to a circle in the period-1 bulb, the nonlinear IFS with such complex mappings is able to generate exotic fractal. Furthermore, if the set of [Formula: see text]-values possesses a specific symmetry in the Mandelbrot set, then the fractal also exhibits the same symmetry. We present a method of generating aesthetic fractals with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text] and with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text].

A continuity result on quadratic matings with respect to parameters of odd denominator rationals

AbstractIn this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

Is regular-closed

For each $n\geq 2$, we investigate a family of iterated function systems which is parameterized by a common contraction ratio $s\in \mathbb{D}^{\times }\equiv \{s\in \mathbb{C}:0&lt;|s|&lt;1\}$ and possesses a rotational symmetry of order $n$. Let ${\mathcal{M}}_{n}$ be the locus of contraction ratio $s$ for which the corresponding self-similar set is connected. The purpose of this paper is to show that ${\mathcal{M}}_{n}$ is regular-closed, that is, $\overline{\text{int}\,{\mathcal{M}}_{n}}={\mathcal{M}}_{n}$ holds for $n\geq 4$. This gives a new result for $n=4$ and a simple geometric proof of the previously known result by Bandt and Hung [Fractal $n$-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for $n\geq 5$.

Control of Dynamic Noise in Transcendental Julia and Mandelbrot Sets by Superior Iteration Method

Researchers and scientists are attracted towards Julia and Mandelbrot sets constantly. They analyzed these sets intensively. Researchers have studied the perturbation in Julia and Mandelbrot sets which is due to different types of noises, but transcendental Julia and Mandelbrot sets remained ignored. The purpose of this article is to study the perturbation in transcendental Julia and Mandelbrot sets. Also, we made an attempt to control the perturbation in transcendental sets by using superior iteration method.

Visualising volumetric fractals

Fractal images have for many years been a rich source of exploration by those in computer science who also have an interest in graphics. They often served as a way of testing the performance of new computing hardware and to explore the capabilities of emerging display technologies. While there have been forays by some into 3D geometric fractals, the 3D equivalents of the Mandelbrot set have been largely ignored. This is largely due to the lack of suitable tools for rendering these sets except perhaps as isosurfaces, a rather unsatisfactory and limited representation. The following will illustrate the application of GPU based raycasting, a now relatively standard approach to volume rendering, to the representation of volumetric fractals. Leveraging existing software that has been designed for general volume visualisation allows the interested 3D fractal explorer to focus on the mathematical generation of the volume data rather than reinventing the entire volume rendering pipeline.

Continued fractions with $SL(2, Z)$-branches: combinatorics and entropy

We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

High‐directivity microstrip antenna with Mandelbrot fractal boundary

High-directivity is a needed feature for small base station antennas in order to provide coverage to a certain geographic area. For wireless communications, arrays of microstrip antennas are frequently used as radiating elements due to their thin profile. In this study, a single microstrip element is proposed to achieve the same directivity as an array with a similar area but without using the complex feeding network usually required. A microstrip antenna inspired by the Mandelbrot fractal is designed achieving a narrow beamwidth in the vertical plane and a wide beamwidth in the horizontal plane, which is especially suitable for hot-spots.

Reduction of dynatomic curves

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$, where $m\geq 2$. We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$. The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$, a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$: one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$, yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $Y_{1}(n)$ for several primes dividing $D_{n}$ when $n=7,8,11$, and $f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

Self-Similarity for the Tricorn

ABSTRACTWe discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author [IK 12], we show that there are many homeomorphic copies of the Mandelbrot sets. With the help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a “baby tricorn” centered at the airplane, hence not a homeomorphism.

Application of discrete dynamic model for the assessment of stability of the world economy development

A discrete dynamic model (DDM) is used to describe the pace of the world GDP annual change rate. The Mandelbrot set of DDM was constructed for different time intervals to assess the ability of the world economic system to maintain a steady pace of development. The article shows that the world economic system is in a fundamentally non-equilibrium state. The Mandelbrot set was proposed to use as a generalized image (pictogram) of the world economy's ability to maintain sustained development rates.

Conformal trace theorem for Julia sets of quadratic polynomials

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

The fractal geometry of Hartree-Fock.

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. In our case, a complex parameter λ controls the strength of the electron-electron interaction. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well.

Estimation of Dynamic Noise in Mandelbrot Map

Julia and Mandelbrot sets have been studied continuously attracting fractal scientists since their creation. As a result, Julia and Mandelbrot sets have been analyzed intensively. In this article, researchers have studied the effect of noise on these sets and analyzed perturbation. Continuing the trend in this article, they analyze perturbation and find the corresponding amount of dynamic noise in the Mandelbrot map. Further, in order to recover a distorted fractal image, a restoration algorithm is presented.

On generalization of Sierpiński gasket in Lobachevskii plane

We construct an analogue of Sierpinski gasket in Lobachevskii plane by means of iterated function system with maps from a transformation group of this space. The investigation of a new family of attractors and a Mandelbrot set associated with it reveals higher capacity of Lobachevskii geometry compared to that of Euclid.

On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

In the paper 'On the dynamics of polynomial-like mappings' Douady and Hubbard introduced the notion of polynomial-like maps. They used it to identify homeomophic copies of the Mandelbrot set inside the Mandelbrot set. They conjectured that in case of primitive copies the homeomorphism between the homeomorphic copy of the Mandelbrot set and the Mandelbrot set is q.-c., and similarly in the satellite case, it is q.-c. off any small neighborhood of the root. These conjectures are now Theorems due to Lyubich. The satellite copies of the Mandelbrot set are clearly not q-c homeomorphic to the Mandelbrot set. But are they mutually q-c homeomorphic? Or even q-c homeomorphic to half of the logistic Mandelbrot set? In this paper we prove that, in general, the induced Douady-Hubbard homeomorphism is not the restriction of a q-c homeomorphism: For any two satellite copies of the Mandelbrot set, the induced Douady-Hubbard homeomorphism is not q-c, if the root multipliers, which are primitive q and q' roots of unity, have q different from q'.