Topology of parameter and dynamical planes

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Parameter plane:

• exterior of M-set = its complement
• M-set:
  • boundary of M-set
  • interior of M-set = hyperbolic components of M-set = critical point in dynamical plane is in periodic orbit

Mandelbrot set is a subset of complex (parameter ) plane :

• connected set
• full set
• compact set

The Mandelbrot set is a compact set it means that it is closed and bounded.
A closed set contains its own boundary = its complement is open.
(Absolute) complement of M-set in C-plane is its exterior.

Hyperbolic component is open set. It does not contain its boundary.
All Misiurewicz points belong to the boundary of the Mandelbrot set

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Sets in dynamical plane:

• basin of attraction of infinite attractor ( infinity ) = D_f(infinity)
• common boundary = (true) Julia set : J(f)
• basin of attraction of finite attractors ( point or cycle)= interior of filled Julia set

K(f) = filled-in Julia set = (true) Julia set + basin of attraction of finite attractors

From The Mandelbrot, Julia and Fatou sets by Evgeny Demidov.
"Points on complex z plane (dynamical or variable space) which under iterations fc for fixed c go to an attractor (attracting fixed point, periodic orbit or infinity) form the Fatou set.
The Julia set (J) is its complement. Therefore the Julia set includes all repelling fixed points, periodic orbits and their preimages."
So C = F(f) + J(f) = K(f) + D_f(infinity)

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• Topology

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Main page

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Autor: adammaj1-at-o2-dot--pl Adam Majewski
Feel free to e-mail me. (:-))
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