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

Local similarity between the Mandelbrot set and Julia sets

Get acquainted with the phenomenon of local 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 current parameter c, which is indicated by the yellow cross in the
left window, belongs to a primitive small Mandelbrot set of period 16 within
the Mandelbrot set M. The filled Julia set K_{c} in the
right window contains a corresponding small Julia set. When it is rescaled
and rotated appropriately, its decorations will look similar to those of the
small Mandelbrot set. Hit the key z repeatedly to zoom into the
parameter plane, and hit p to rescale the dynamic plane.

You may either hit z many times before hitting p, or hit
z and p alternately to zoom into both planes simulataneously.
Hit v to zoom out again. (If you hit h to restart, you must
use x afterwards before hitting p again.)

When you have zoomed in far enough for the small Mandelbrot set to become
prominent, you may change the parameter c by clicking into the left
window, or move it with the arrow keys. Hit p to rescale the
new dynamic plane. Although the small Julia set is changing, its rescaled
decorations remain unchanged except for bending. (Try to hit p
immeadiately after hitting the arrow key.)

This kind of local similarity happens for many primitive small Mandelbrot
sets, which are found everywhere in the antennae of M. First set the
current parameter by clicking inside its cardioid, hit x and
Enter to accept the period shown, which moves the parameter to the
center. Afterwards you may hit p to rescale the dynamic plane
according to local similarity at the specified center, zoom out with
v and in again with z, or change the parameter before hitting
p. Note that the period will be cleared when you hit h or use
any command referring to asymptotic similarity: then p will use the
latest Misiurewicz point for rescaling.

Web application under construction ...

Mathematical explanation under construction ...
See demo 7 of Mandel and the
presentation from the workshop
in complex dynamics, Søminestationen in Holbæk, October 2007.