Construction of 3D Mandelbrot Set and Julia Set

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

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Complex Mapping of 3DCG Models with Julia Set and Mandelbrot Set

Sixth International Conference On Advances in Computing, Electronics and Electrical Technology - CEET 2016 ◽

10.15224/978-1-63248-109-2-25 ◽

2016 ◽

Author(s):

FUKO NAKAGAMI ◽

NINGPING SUN

Keyword(s):

Julia Set ◽

Mandelbrot Set ◽

Complex Mapping

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Multicorns are not path connected

Frontiers in Complex Dynamics ◽

10.23943/princeton/9780691159294.003.0006 ◽

2014 ◽

Cited By ~ 1

Author(s):

John Hamal Hubbard ◽

Dierk Schleicher

Keyword(s):

Julia Set ◽

Mandelbrot Set ◽

Quadratic Polynomials ◽

Unicritical Polynomials ◽

Path Connected ◽

Special Case ◽

Connectedness Locus

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

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Conformal trace theorem for Julia sets of quadratic polynomials

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.124 ◽

2017 ◽

Vol 39 (9) ◽

pp. 2481-2506 ◽

Cited By ~ 6

Author(s):

A. CONNES ◽

E. MCDONALD ◽

F. SUKOCHEV ◽

D. ZANIN

Keyword(s):

Hausdorff Dimension ◽

Jordan Curve ◽

Hausdorff Measure ◽

Julia Set ◽

Julia Sets ◽

Mandelbrot Set ◽

Trace Theorem ◽

Quadratic Polynomials ◽

Full Proof

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.

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Non-Differentiable Structure in the Generalized Mandelbrot Set

Zeitschrift für Naturforschung A ◽

10.1515/zna-1988-0319 ◽

1988 ◽

Vol 43 (3) ◽

pp. 287-288

Author(s):

J. Peinke ◽

J. Parisi ◽

B. Röhricht ◽

O. E. Rössler ◽

W. Metzler

Keyword(s):

Vortex Structure ◽

Julia Set ◽

Mandelbrot Set ◽

Fractal Structures ◽

Differentiable Structure ◽

Self Similar

Abstract The generalized Mandelbrot set, described previously, contains - like the original Mandelbrot set - non-differentiable self-similar fractal structures. An example looking like a vortex structure is presented along with a corresponding generalized Julia set.

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Describing quadratic Cremer point polynomials by parabolic perturbations

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798108301 ◽

1998 ◽

Vol 18 (3) ◽

pp. 739-758 ◽

Cited By ~ 4

Author(s):

DAN ERIK KRARUP SØRENSEN

Keyword(s):

Infinite Order ◽

Julia Set ◽

Main Idea ◽

Mandelbrot Set ◽

Basic Tool ◽

Quadratic Polynomials ◽

Hyperbolic Component ◽

External Rays ◽

External Ray ◽

Single Comb

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

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Accumulation theorems for quadratic polynomials

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000897x ◽

1996 ◽

Vol 16 (3) ◽

pp. 555-590 ◽

Cited By ~ 1

Author(s):

Dan Erik Krarup Sørensen

Keyword(s):

Julia Set ◽

Julia Sets ◽

Mandelbrot Set ◽

Quadratic Polynomials ◽

Hyperbolic Component ◽

Non Local ◽

External Ray ◽

The One ◽

Image Position ◽

Symmetrical Points

AbstractWe consider the one-parameter family of quadratic polynomials:i.e. monic centered quadratic polynomials with an indifferent fixed point αtand prefixed point −αt. LetAt, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to aGδ-dense subset ofthere is an explicitly given external ray accumulating onAt. In the caseAt= {±αt} the theorem is known as theDouady accumulation theorem.Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the denseGδsets arising from the first two possible choices ofAtare the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αtand –αtmay be joined by an arc in the Julia set.

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DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP

Modern Physics Letters B ◽

10.1142/s0217984908014717 ◽

2008 ◽

Vol 22 (04) ◽

pp. 243-262 ◽

Cited By ~ 1

Author(s):

XINGYUAN WANG ◽

QINGYONG LIANG ◽

JUAN MENG

Keyword(s):

Analytic Function ◽

Julia Set ◽

Julia Sets ◽

Mandelbrot Set ◽

Quantitative Criterion ◽

Boundary Equation ◽

Computer Aided ◽

Definition Of ◽

Mathematics Method ◽

Periodic Bifurcation

The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

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THE QUASI-SINE FIBONACCI HYPERBOLIC DYNAMIC SYSTEM

Fractals ◽

10.1142/s0218348x10004725 ◽

2010 ◽

Vol 18 (01) ◽

pp. 45-51 ◽

Cited By ~ 6

Author(s):

XING-YUAN WANG ◽

FENG-DAN GE

Keyword(s):

Dynamic System ◽

Fixed Points ◽

Critical Point ◽

Dynamic Behavior ◽

Real Axis ◽

Critical Points ◽

Julia Set ◽

Mandelbrot Set ◽

Escape Time ◽

Fibonacci Function

This paper researches the dynamic behavior of a general form of the Fibonacci function, which is a quasi-sine Fibonacci function. It analyses the fixed points of the quasi-sine Fibonacci function on the real axis and the complex plane, and then constructs the Julia set of it using the escape-time method, discovering that the Julia set is fractal and it is on the x-axis symmetry. Using the conception of critical point, the quasi-sine Fibonacci function is generalized. Later the paper examines the dynamic behavior of the generalized quasi-sine Fibonacci function on critical points, and finds that the Mandelbrot set is also on the x-axis symmetry. Finally, it is discovered that there is a jumping phenomenon on the critical points.

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Total disconnectedness of Julia sets of random quadratic polynomials

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.148 ◽

2021 ◽

pp. 1-17

Author(s):

KRZYSZTOF LECH ◽

ANNA ZDUNIK

Keyword(s):

Dynamical System ◽

Julia Set ◽

Julia Sets ◽

Mandelbrot Set ◽

Quadratic Polynomials ◽

Large Disc ◽

Fatou Sets ◽

Autonomous Version ◽

Composition Of Functions ◽

Totally Disconnected

Abstract For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

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