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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 for rotational number 1/34 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

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  • $\begingroup$ 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). $\endgroup$
    – Adam
    Commented Aug 18, 2013 at 14:47
  • $\begingroup$ 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. $\endgroup$
    – Sheldon L
    Commented Aug 28, 2013 at 19:37

1 Answer 1

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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.

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

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