Mandelbrot Set Research Articles

Fine structure of connectedness loci

We study distributions of the harmonic and Lebesgue measures on the boundary of the connectedness locus \(\mathcal{M}_{d}\) for uni-critical polynomials \(z^{d}+c\) and dynamical properties which are typical with respect to the harmonic measure. One result is a conformal similarity between \(\mathcal{M}_{d}\) and the corresponding Julia set for almost every point of \(\mathcal{M}_{d}\) with respect to the harmonic measure. It is also shown that the harmonic measure is supported on Lebesgue density points of the complement of \(\mathcal{M}_{d}\) which are not accessible from outside within John angles and at which the boundary of \(\mathcal{M}_{d}\) ‘spirals’ infinitely often in both directions. In the case of quadratic polynomials, we prove that every parameter c from the boundary of the Mandelbrot set \(\mathcal{M}\) that is recurrent under iterates of \(z^{2}+c\) but not infinitely tunable is a Lebesgue weak density point of the complement of \(\mathcal{M}\). A direct consequence of our theory is the Yoccoz local connectivity theorem. A new technical ingredient in the paper is the method of “amplification” which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding Julia set. The pointwise conformality of the similarity map is derived through inducing techniques and a classical TWB-theory.

Internal Addresses of the Mandelbrot Set and Galois Groups of Polynomials

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a convenient and efficient way of describing the combinatorial structure of the Mandelbrot set, and of giving geometric meaning to the ubiquitous kneading sequences in human-readable form (Sects. 3 and 4). A simple extension, angled internal addresses, distinguishes combinatorial classes of the Mandelbrot set and in particular distinguishes hyperbolic components in a concise and dynamically meaningful way. This combinatorial description of the Mandelbrot set makes it possible to derive existence theorems for certain kneading sequences and internal addresses in the Mandelbrot set (Sect. 6) and to give an explicit description of the associated parameters. These in turn help to establish some algebraic results about permutations of periodic points and to determine Galois groups of certain polynomials (Sect. 7). Through internal addresses, various areas of mathematics are thus related in this manuscript, including symbolic dynamics and permutations, combinatorics of the Mandelbrot set, and Galois groups.

The parameter space of cubic laminations with a fixed critical leaf

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

Roots, Schottky semigroups, and a proof of Bandt’s conjecture

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’${\mathcal{M}}$for pairs of similarities: this is the set of complex numbers$z$with$0<|z|<1$for which the limit set of the semigroup generated by the similarities$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$is connected. Equivalently,${\mathcal{M}}$is the closure of the set of roots of polynomials with coefficients in$\{-1,0,1\}$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ‘holes’ in${\mathcal{M}}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ‘exotic’ components of the space of (2-generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique oftrapsto construct and certify interior points of${\mathcal{M}}$, and use them to prove Bandt’s conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in${\mathcal{M}}$.

The Decoration Theorem for Mandelbrot and Multibrot Sets

We prove the decoration theorem for the Mandelbrot (and Multibrot sets) which says that when a little Mandelbrot set is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.

Moiré interferences in the map of orbits of the Mandelbrot Set

This article shows the presence of Moiré Interference patterns in the map of periods of the Mandelbrot Set. It describes the requirements for their appearance and shows that such interferences are highly sensitive to the original conditions that define their calculation. The specific case herein studied shows that the Moiré interference patterns appearing in a picture of a section of the map of orbits are unpredictable, even if we obtain different maps from very similar original conditions. It begins with a brief description of the Mandelbrot Set and some of the characteristics of its orbits, the Moiré Patterns, as well as a concise introduction to a description of the Discrete Wavelet Transform. In order to develop the proposed specific case, a Multi-resolution Analysis method based on the Discrete Wavelet Transform has been used. It is significant that Moiré Interference Patterns always appear when the order of magnitudes reaches a certain limit where, what is considered as hypothetically continuous, behaves as a discrete pattern. The patterns as shown by the Wavelet analysis change drastically at the slightest modification in the original calculation conditions and it does not seem possible to predict their shape beforehand. This article ends with some conclusions and suggestions.

Scalable Loop Self-Scheduling Schemes for Large-Scale Clusters and Cloud Systems

Cloud systems have demonstrated the powerful computation and storage capability in many scientific applications. In this paper, we propose a class of scalable distributed loop self-scheduling schemes to achieve good load balancing and scalability. We implemented these schemes on a large-scale cluster and on a heterogeneous cloud system. The schemes consider the distribution of the output data, which can help reduce communication overhead and improve scalability. We evaluated the schemes using four scientific computations: Mandelbrot set, adjoint convolution, matrix multiplication and quick sort. The results show that the new schemes achieve better load balancing, better scalability and better overall performance than standard distributed loop self-scheduling schemes.

NEW JULIA AND MANDELBROT SETS FOR A NEW FASTER ITERATIVE PROCESS

Fixed point iterative procedures are the backbones of fractal geometry. In existing literature Julia sets, Mandelbrot sets and their variants have been studied using one - step, two - step, three - step and four - step iterative process. Recently, M. Abbas and T. Nazir (12) introduced a new iterative process (a four-step iterative process) which is faster than all of Picard, Mann and Agarwal processes. In this paper, we obtain further generalizations of Julia and Mandelbrot sets using this faster iterative process for quadratic, cubic and higher degree polynomials. Further, we analyze that few Julia and Mandelbrot sets took the shape of Lord Ganesha (name of Hindu God), Dragon and Urn.

Exponential Sensitivity and its Cost in Quantum Physics.

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.

Symbolic template iterations of complex quadratic maps

The behavior of orbits for iterated polynomials has been widely studied since the dawn of discrete dynamics as a research field, in particular in the context of the complex quadratic family $$f :\mathbb {C} \rightarrow \mathbb {C}$$ , parametrized as $$f_c(z) = z^2 + c$$ , with $$c \in \mathbb {C}$$ . While more recent research has been studying the orbit behavior when the map changes along with the iterations, many aspects of non-autonomous discrete dynamics remain largely unexplored. Our work is focused on studying the behavior of pairs of quadratic maps (1) when iterated according to a rule prescribed by a binary template and (2) when the maps are organized as nodes in a network, and interact in a time-dependent fashion. We investigate how the traditional theory changes in these cases, illustrating in particular how the hardwired structure (the symbolic template, and respectively the adjacency graph) can affect dynamics (behavior of orbits, topology of Julia and Mandelbrot sets). Our current manuscript addresses the first topic, while the second topic is the subject of a subsequent paper. This is of potential interest to a variety of applications (including genetic and neural coding), since (1) it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the outcome and (2) it relates to algorithms of synaptic restructuring and neural dynamics in brain networks.

Control of the spatial Mandelbrot set generated in coupled map lattice

This paper broadens the research on planar Mandelbrot set to the spatial case by investigating the Mandelbrot set generated in coupled map lattice (CML M-set for short). The visualization and symmetry property of the 3D slice of CML M-set are investigated. Furthermore, a proper controller is designed for the 3D slice of CML M-set to achieve its symmetry change, position change and size change. Numerical simulations are included to verify the effectiveness of the control method.

Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

This work explores the Julia and Mandelbrot sets of the Gamma function by extending the function to the entire complex plane through analytic continuation and functional equations. Various Julia and Mandelbrot sets associated with the Gamma function are generated using the iterative function , with different parameter values. To produce an accurate result using the integral definition of the Gamma function, a large number of terms would have to be added during the numerical integration procedure; this makes computation of Gamma function a very difficult task. To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision is used. The resulting images reveal that the fractal (chaotic) behaviour found in elementary functions is also found in the Gamma function. The chaotic nature of the Julia and Mandelbrot sets provides a way of understanding complexity in systems as well as just in shapes.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials

Themulticornsare the connectedness loci of unicritical antiholomorphic polynomials$\bar{z}^{d}+c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

Mandelbrot fractal microstrip antennas

This article proposes the use of complex plane fractal Mandelbrot set geometries, which are of the escape-time fractal type, in the development of microstrip patch antennas fed by a coaxial cable. These antennas are developed for near millimeter-wave applications. Basically, the Mandelbrot fractal geometry is designed from a circular geometry enabling intrinsically a patch element size reduction. In addition, this work is aimed to investigate the effect of using Mandelbrot fractal patch geometry with different iterations. Thereafter, Mandelbrot microstrip patch antennas are designed and simulated using Ansoft HFSS software package. Prototypes are fabricated and tested to validate the design procedure. Good agreement is obtained between simulation and experimental results. © 2016 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:83–86, 2016

Colored HOMFLY and generalized Mandelbrot set

Mandelbrot set is a closure of the set of zeroes of $resultant_x(F_n,F_m)$ for iterated maps $F_n(x)=f^{\circ n}(x)-x$ in the moduli space of maps $f(x)$. The wonderful fact is that for a given $n$ all zeroes are not chaotically scattered around the moduli space, but lie on smooth curves, with just a few cusps, located at zeroes of $discriminant_x(F_n)$. We call this phenomenon the Mandelbrot property. If approached by the cabling method, symmetrically-colored HOMFLY polynomials $H^{\cal K}_n(A|q)$ can be considered as linear forms on the $n$-th "power" of the knot ${\cal K}$, and one can wonder if zeroes of $resultant_{q^2}(H_n,H_m)$ can also possess the Mandelbrot property. We present and discuss such resultant-zeroes patterns in the complex-$A$ plane. Though $A$ is hardly an adequate parameter to describe the moduli space of knots, the Mandelbrot-like structure is clearly seen -- in full accord with the vision of arXiv:hep-th/0501235, that concrete slicing of the Universal Mandelbrot set is not essential for revealing its structure.

A Mandelpinski maze for rational maps of the form [formula omitted

In this paper we identify a new type of structure that lies in the parameter plane of the family of maps zn+λ/zd where n≥2 is even but d≥3 is odd. We call this structure a Mandelbrot–Sierpinski maze. Basically, the maze consists at the first level of an infinite string of alternating Mandelbrot sets and Sierpinski holes that lie along an arc in the parameter plane for this family. At the next level, there are infinitely many smaller Mandelbrot sets and Sierpinski holes that alternate on the arc between each Mandelbrot set and Sierpinski hole on the previous level, and then finitely many other Mandelbrot sets and Sierpinski holes that extend away from the given Mandelbrot set in a pair of different directions. And then this structure repeats inductively to produce the “Mandelpinski” maze.

Non-landing parameter rays of the multicorns

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of $\mathcal{M}_d^*$, and the Misiurewicz parameters are not dense on the boundary of $\mathcal{M}_d^*$.

Mandelpinski spokes in the parameter planes of rational maps

In this paper we describe a new structure that arises in the parameter plane of the family of maps where is even but is odd. We call these structures Mandelbrot–Sierpiński spokes (or, for short, ‘Mandelpinski spokes’). It is known that there are infinitely many baby Mandelbrot sets in these parameter planes that are part of what is called the Mandelpinski maze for these maps. We show here that there are infinitely many ‘spokes’ emanating from each of these Mandelbrot sets. Each spoke consists of infinitely many alternating Mandelbrot sets and Sierpiński holes that lie along a certain arc that tends away from the given Mandelbrot set in a certain direction.

Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps

We study the dynamics of the family of rational maps of the form $$\begin{aligned} f_{d,{\uplambda }}(z)={\uplambda }\left( z+\frac{1}{z^{d-1}}\right) , \quad d \ge 3, \quad {\uplambda }\in \mathbb C{\setminus } \{0\}. \end{aligned}$$ Among other things, we show that the parameter planes for these maps contain infinitely many copies of the Mandelbrot set as well as infinitely many “blowup points”, i.e., parameters for which the critical orbits map to \(\infty \), so the Julia set is the entire sphere. Our efforts are aided by the useful observation that for fixed \(d \ge 3\), this family is conformally conjugate on the entire Riemann sphere to the family of relaxed Newton maps for \(p_d(z) = z^d-1\). The conjugacy allows us to move from one family to the other in order to find simpler proofs of our results, as well as establishing a dictionary of results from one family to the other.