Wolf Jung jung@mndynamics.com
Rombachstrasse 99, 52078 Aachen, Germany.

Asymptotic self-similarity of the Mandelbrot set

Get acquainted with the phenomena of asymptotic scaling and similarity
in an interactive session with the Java applet Juliette below. Click the
gray column to give it keyboard focus (do not click the images). Use the
following commands or the buttons below:

The yellow cross in the left window is indicating the current parameter
c=a, which is a Misiurewicz point of preperiod 3 and period 1.
The critical value z=a is mapped to the fixed point
α_{a} in 3 iterations of f_{a}(z). You
may follow the orbit in the right window by hitting the key f
a few times.

Hit t repeatedly to rescale the parameter plane around
c=a with the complex scaling factor
ρ=f_{a}'(α_{a})=2α_{a}.
The image is enlarged by |ρ| and rotated by the argument of ρ, which
is approximately 120^{o}. The rescaled images show that
ρ^{n}(M-a) is converging to an asymptotic
model set: M is asymptotically self-similar at c=a.

The Julia set K_{a} in the right window is asymptotically
self-similar as well, and both sets are asymptotically similar to each other:
(K_{a}-a)≈λ(M-a). Hit
p to see the subset of K_{a} corresponding to the
subset of M. Hit a to restart.

There is a sequence of small Mandelbrot sets spiraling towards a. The
centers c_{n} are labeled such that the period is n+7.
Hit m to see period 7, and hit t repeatedly to increase the
period. After n steps, the distance to a is reduced by the
factor ρ^{n}, and the image is rescaled with
ρ^{2n}. The small Mandelbrot sets are converging to a
copy of M, and the decorations are getting longer and thinner.

There are asymptotic models for
ρ^{γn}(M-c_{n}) on the levels
γ=1, γ=3/2, γ=7/4, γ=15/8 ... Hit 0, 1,
2, or 3 to choose the level, and hit t repeatedly to
increase the order n. Hit p to see corresponding subsets of
K_{c}.

To choose any Misiurewicz point a, set the current point close to it,
hit x, and enter the preperiod and period. The commands a,
t, and p refer to the new Misiurewicz point afterwards. To
enable the levels 0, 1, 2, 3, and m,
you must set the current parameter inside the hyperbolic component around the
center c_{0} before hitting x. E.g., hit i and
paste “-0.04421 0.98658” for a component of period 5. Then hit x and enter “1,2” to find a Misiurewicz point of preperiod
1 and period 2, which is a=i.

Web application under construction ...

Mathematical explanation under construction ...
See demo 6 of Mandel.