Parabollic fixed points and Siegel disk
If multiplicator of a fixed point satisfies
|λ| ≠ 1 then dynamics in its vicinity
is simple attraction or repelling.
It is much more complex for |λ| = 1.
For λn = 1
(i.e. λ = e2π i m/n
where m, n are integer),
the fixed poit is called parabollic.
E.g. as we have seen before,
for c → 1/4 attracting and repelling fixed points
merge together and make parabollic fixed point
with multiplicator λ = 1 (in the center of the white
square below). The point has one attracting and one repellinga directions.
You can see that white points of the orbit starting at z = 0 are
attracted to this parabollic point. Points to the right from
the fixed point go to infinity.
Parabollic point with 3 attracting and repelling directions
is shown below.
If λ = e2π i x (where x is
real irrational number) then in vicinity of fixed point
for "almost all" x the mapping is equivalent to
rotation by 2π x angle.
For some x values the map is not equivalent to rotation.
These fixed points are called Cremer points.
It is impossible to use irrational numbers on computers
but mathematitiens have strong proof that Siegel and
Cremer points exist .
To the left you see that orbit starting at z = -0.2 + 0.2i
rotates around fixed point. It is very likely that we have
Siegel disk for this C value (at least for 1500 iterations :)
 John Milnor "Dynamics in One Complex Variable"
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updated 5 Sep 2015