The Mandelbrot Set: Ordering the Julia Sets

1992 ◽  
pp. 415-473
Author(s):  
Heinz-Otto Peitgen ◽  
Hartmut Jürgens ◽  
Dietmar Saupe
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set

Julia Sets and the Mandelbrot Set

1986 ◽  
pp. 161-174 ◽  
Author(s):  
Adrien Douady
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set

The Mandelbrot Set: Ordering the Julia Sets

2004 ◽  
pp. 783-837
Author(s):  
Heinz-Otto Peitgen ◽  
Hartmut Jürgens ◽  
Dietmar Saupe
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set

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.

Similarity between the Dynamic and Parameter Spaces in Cubic Mappings

Fractals ◽  
1998 ◽  
Vol 06 (03) ◽  
pp. 293-299
Author(s):  
Chia-Chin Cheng ◽  
Sy-Sang Liaw
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set ◽  
Fractal Structures ◽  
Parameter Spaces

We have extended the work of Lei Tan on the similarity between the Mandelbrot set and the Julia sets. We show that the fractal structures of dynamic and parameter spaces are asymtotically similar at Misiurewicz points for the cubic mappings.

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.

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