A GENERALIZED MANDELBROT SET FOR BICOMPLEX NUMBERS

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 355-368 ◽  
Author(s):  
DOMINIC ROCHON
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Bicomplex Numbers ◽  
Set Of Points

We use a commutative generalization of complex numbers called bicomplex numbers to introduce bicomplex dynamics. In particular, we give a generalization of the Mandelbrot set and of the "filled-Julia" sets in dimensions three and four. Also, we establish that our version of the Mandelbrot set with quadratic polynomial in bicomplex numbers of the form w2 + c is identically the set of points where the associated generalized "filled-Julia" set is connected. Moreover, we prove that our generalized Mandelbrot set of dimension four is connected.

Julia sets converging to filled quadratic Julia sets

2012 ◽  
Vol 34 (1) ◽  
pp. 171-184 ◽  
Author(s):  
ROBERT T. KOZMA ◽  
ROBERT L. DEVANEY
Keyword(s):  
Singular Perturbations ◽  
Julia Set ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Hausdorff Topology ◽  
Quadratic Map ◽  
Hyperbolic Component

AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map

1994 ◽  
Vol 14 (4) ◽  
pp. 787-805 ◽  
Author(s):  
Karsten Keller
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Topological Classification ◽  
Closed Curves ◽  
Abstract Description ◽  
Kneading Sequence ◽  
Doubling Map ◽  
Sturmian Sequences ◽  
Set Of Points

AbstractBy the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling maphon the circleT. In particular, a connected and locally connected Julia set can be considered as a topological factorT/ ≈ ofTwith respect to a specialh-invariant equivalence relation ≈ onT, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α →αfromTonto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia setsT/α. It turns out thatT/αcontains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.

ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES

Fractals ◽  
2009 ◽  
Vol 17 (03) ◽  
pp. 241-255 ◽  
Author(s):  
V. GARANT–PELLETIER ◽  
D. ROCHON
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Mandelbrot Set ◽  
Topological Characterization ◽  
Multicomplex Numbers ◽  
3D Fractals

In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

scholarly journals Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

2019 ◽  
Vol 3 (1) ◽  
pp. 6 ◽  
Author(s):  
Vance Blankers ◽  
Tristan Rendfrey ◽  
Aaron Shukert ◽  
Patrick Shipman
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Hyperbolic Plane ◽  
Julia Sets ◽  
Number System ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Fractal Sets ◽  
Mandelbrot Sets

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
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.

ON A BICOMPLEX DISTANCE ESTIMATION FOR THE TETRABROT

2005 ◽  
Vol 15 (09) ◽  
pp. 3039-3050 ◽  
Author(s):  
ÉTIENNE MARTINEAU ◽  
DOMINIC ROCHON
Keyword(s):  
Fractal Structure ◽  
Julia Sets ◽  
Distance Estimation ◽  
Mandelbrot Set ◽  
Simple Method ◽  
Bicomplex Numbers ◽  
3D Fractals

In this article, we present some distance estimation formulas that can be used to ray traced slices of the bicomplex Mandelbrot set and the bicomplex filled-Julia sets in dimension three. We also present a simple method to explore and infinitely approach these 3D fractals. Because of its rich fractal structure and symmetry, we emphasize our work on the generalized Mandelbrot set for bicomplex numbers in dimension three: the Tetrabrot.

Accumulation theorems for quadratic polynomials

1996 ◽  
Vol 16 (3) ◽  
pp. 555-590 ◽  
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.

DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP

2008 ◽  
Vol 22 (04) ◽  
pp. 243-262 ◽  
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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