To compute |φ(z)| at a point z we must simply follow the
orbit z_{k} = f_{c}^{ok}(z) until at a step
k we reach some point which is very far from 0 where
f_{c} is close to f_{o} (and z_{k}
to w_{k}). Then we make k itterations
f_{o}^{-1} (in the "reverse" direction) i.e.
|φ(z)| = lim_{k→∞}
|f^{ ok}(z)|^{1/2k}.
G(z) = lim_{n→∞ }ln|f^{ on}(z)|
/ 2^{n}.
This implies also that G(f(z)) = 2 G(z). It is evident that far from charged body equipotential curves are circles. Let R is a large circle then G(f ^{-ok}(R)) = G(R)/2^{k} = const therefore preimages of the circle are equipotential curves for the Julia set (see the picture to the left for R = 10). |
Electric field lines (orthogonal to equipotential curves) are called
external rays. They are preimages of the field lines of charged circle
r exp(2πit), r>1. Each such ray R_{t}
can be specified by its angle at infinity t.
Due to isomorphism φ(z) external rays dynamics under f
is the same as dynamics of the straight rays under z^{2}. E.g.
f(R_{t }) = R_{2t }. To the left every point is painted in the red, green and blue colors if for r > 10 its orbit falls into the sectors (0^{o}, 120^{o}), (120^{o}, 240^{o}), (240^{o}, 360^{o}) correspondigly. With enough patience you can see equipotential curves and orthogonal field lines. Exteral rays with rational angles t = 0, 1/4, 1/3, 1/2... are marked here. |
At last to the left external rays p/12 for p = 0,2,..11
are plotted by means of the
Wolf Jung's algorithm.
As since I used only simplest Jung's magic this applet is a bit
buggy for large q.
If you need flexible and accurate program to explore external rays and much
more use his Mandel.
I am grateful to Adam Majewski for he convinced me to translate Wolf Jung's sources into Java. |
In particular, if we choose a constant G_{o} > G(0), then G(z) = G_{o} is a simple closed curve, canonically parametrized by the angle of the corresponding dynamic ray. In particular, the critical value z = f_{c}(0) = c has a well defined external angle, which we denote by t(c). However, for G_{o} = G(0) this points G(z) = G(0) make a figure eight curve (see to the left). The open set G(z) < G(0) splits as a disjoint union U_{o} and U_{1}, where the U_{b} are the regions enclosed by the two lobes of this figure eight. |
According to Douady and Hubbard ψ_{c} is a conformal
isomorphism from the complement of M onto the complement of the closed
unit disk. Therefore it determines equipotential curves and external rays for
the Mandelbrot set.
It is known that an external ray whose angle is rational actually lands on
M. That is |
[1] J. Milnor Dynamics in One Complex Variable: Introductory Lectures. Preprint ims90-5