What is the shape of external rays landing on fixed points in case of quadratic discrete dynamical system?

Question

In case of parabolic discrete dynamical system based on the complex quadratic polynomial

fc(z) = z^2 + c

some external rays land on alfa fixed point. Hera are 34 external rays landing on fixed point for rotational number 1/34.

c = 0.258368 + 0.001564*i

1. It seems that these rays spiral around fixed point. Is it true or it is only some numerical error ?

2. What can be said about shape of these external rays in case of rotational numer 1/n ?

3. how can I compute points on external ray near parabolic fixed point ?

TIA

Answer 1

In Complex Dynamics by Carleson and Gamelin page 40 ther is a : "The gap between 2 consecutive petals is contained in a cusp bounded by curves with :

$|\theta - \theta'_k| \sim |z|^{1/p}$

"

It should describe shape of these rays.