Local dynamics at a fixed point
We can write a complex multiplier λ
(in the polar coordinate system) as
λ = ρ exp(iφ ).
Then iterations (or images) of a point (z_{o} +
ε ) in the vicinity of a fixed point
z_{o} = f(z_{o}) are
z_{k} = f^{ ok}(z_{o} +
ε ) = z_{o} +
λ ^{k}ε + O(ε^{ 2}) ~ z_{o} +
ρ^{ k} e^{ikφ}ε .
That is, if we put coordinate origin to z_{o} , after
every iteration point z_{k+1} is rotated by angle
φ with respect to the previous position
z_{k} and its radius is scaled by ρ
= λ.
For φ = 2π
m/n points z_{k} jump exactly m rays in the
counterclockwise direction at each iteration and make nrays "star"
or "petals" structures discussed on the previous page.
These structures are more "visible" for
ρ = 1 + δ , δ  << 1 (e.g. near the main cardioid
border).
Attracting fixed point
For ρ < 1 all points in the vicinity of
attractor z_{o} move smoothly to z_{o} .
You can see "star" structures made by orbit of the critical point.
Repelling fixed point

For c outside the main cardioid, ρ > 1 and the fixed
point z_{o} becomes repelling (and
it lies in J). Connected J set separates basin of attracting cycle
and basin of infinit point. Therefore in the vicinity of z_{o}
rotations by 2π m/n generate npetals
structures made of these two basins. Points in petals are attracted by
periodic cycle and points in narrow whiskers go to infinity.



You see below, that rotational symmetry near repeller z_{o}
keeps for "dendrite" and Cantor dust Jsets too.
Spiral structures in the Julia sets
It is evident, that if φ = 2π m/n + δ ,
then mapping
z_{k} = f^{ ok}(z_{*} +
ε ~ z_{*} + ρ^{ k}
e^{ikφ}ε
generates spiral structures in the neighbourhood of the fixed point
z_{*} . Some of these spirals are shown below.
Next we can investigate stability of fixed points and period 2 orbit
of quadratic mappings analytically.
Contents
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Next: Attracting fixed point and period 2 orbit
updated 17 August 2003