Julia Sets
The Julia set of a quadratic function
is the boundary of the keep set
or the boundary of the escape set
and separates the points which escape to infinity under repeated iteration by
and those which don't. The Julia set contains all the periodic repelling points ofand is the smallest closed set which contains all the repelling periodic periodic points of f. Two are illustrated below.
The Julia set has the following properties:
For each
the Julia set
is a non empty compact subset of
- is completely invariant.
- is symmetric under rotation by
about 0.
The Julia set is often plotted using the following method. If
then the solutions of
also lie in
The solution are
so backwards iteration gives two new points of
and
iterating backwards again with these two points gives four new points
and
Iterating backwards repeatedly gives more points. It can be shown that the Julia set is the smallest closed set containing all the backwards iterates of any given point ofBecause of the amount of calculation involved, not all the backwards iterates are calculated at each stage. A single iterate from each stage may be randomly selected at the backwards iterates found for that point.