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 (zo +
ε ) in the vicinity of a fixed point
zo = f(zo) are
zk = f ok(zo +
ε ) = zo +
λ kε + O(ε 2) ~ zo +
ρ k eikφε .
That is, if we put coordinate origin to zo , after
every iteration point zk+1 is rotated by angle
φ with respect to the previous position
zk and its radius is scaled by ρ
For φ = 2π
m/n points zk jump exactly m rays in the
counterclockwise direction at each iteration and make n-rays "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
Attracting fixed point
For ρ < 1 all points in the vicinity of
attractor zo move smoothly to zo .
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 zo 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 zo
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 zo
keeps for "dendrite" and Cantor dust J-sets too.
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Next: Spiral structures in the Julia sets
updated 12 Sep 2013