Local dynamics at a fixed point
We can write a complex multiplier λ
(in the polar coordinate system) as
λ = |λ| exp(iφ ).
Then iterations (or images) of a point (zo +
ε ) in the vicinity of a fixed point
z∗ = f(z∗ ) are
zk = f ok(z∗ +
ε ) = z∗ +
λ kε + O(ε 2) ~ z∗ +
|λ| k eikφε.
That is, if we put coordinate origin to z∗ , after
every iteration point zk+1 is rotated by angle
φ with respect to the previous position
zk and its radius is scaled by |λ|.
For φ = 2π m/n points zk
jump exactly m rays in the
counter-clockwise direction at each iteration and make n-rays "star"
or "petals" structures. These structures are more "visible" for
λ = 1 + δ , |δ | << 1 (e.g. near the main cardioid
Attracting fixed point
For |λ| < 1 all points in the vicinity of
attractor z∗ move smoothly to z∗ .
You can see "star" structures made by orbit of the critical point.
Repelling fixed point
For c outside the main cardioid, |λ| > 1 and the fixed
point z∗ becomes repelling
(it lies in J). Connected J set separates basin of attracting cycle
and basin of infinite point. Therefore in the vicinity of z∗
rotations by 2π m/n generate n-petals
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∗
keeps for "dendrite" and Cantor dust J-sets too.
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updated 12 Sep 2013