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

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 only to the first question: Here c = 0.258368 + 0.001564*i and it is not a parabolic parameter. Here rotational number = 0.02941078600973229. It should be 1/34=0.02941176470588235 and the difference = 9.786961500633795e-7. So the image is good ( rays near alfa are approximated by straight line) but it is not parabolic case).
– Adam
Commented Aug 18, 2013 at 14:47
• I'm very much still learning "landing rays". I got the 34 Landing rays to work for the Julia for n=34 parabolic point, which is very close to your "c". It works better for C=0.258385930 + 0.00148483651i, which is the hyperbolic center. I used Wolf Jung's Mandel program to play external rays, plotting 1/(2^34-1),2/(2^34-1)....2^33/(2^34-1) for the angles. It looks more or less like your picture. Again, still learning... The parabolic case is going to have extremely slow convergence, so I would imagine the rays would need to get very very close to the unit circle. Hyperbolic case easier. Commented Aug 28, 2013 at 19:37

## 1 Answer

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.

• I need to go back and reread Carleson & Gamelin's book, and see if I can understand more now. Commented Sep 24, 2013 at 20:47