4.2: Local dynamics at a fixed point

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We can write a complex multiplier λ (in the polar coordinate system) as

λ = |λ| exp(iφ ).

Then iterations (or images) of a point (z_o + ε ) in the vicinity of a fixed point z_∗ = f(z_∗) are

z_k = f^ok(z_∗ + ε ) = z_∗ + λ ^kε + O(ε²) ~ z_∗ + |λ|^k e^ikφε.

That is, if we put coordinate origin to z_∗, 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 |λ|.

For φ = 2π m/n points z_k 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 border).

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.