Fractals/Iterations in the complex plane/cremer

From Wikibooks, open books for an open world
Jump to: navigation, search
Topologist's single comb
The intricated double comb for r=3/4
  • there are Cremer Julia set for quadratic polynomial,
  • there are no images of such Julia sets
  • a Cremer Julia set has no interior. It contains single or double comb. [1] "Objects such as combs (homeomorphs of the

product of a Cantor set and an interval) had been expected to be found within hedgehogs." (Kingshook Biswas [2]

  • Cremer Julia sets are not locally connected [3][4]


Parameter[edit]

It is hard to give values of Cremer parameters ( c-points).

"The Cremer parameters are on the boundaries of hyperbolic components at specific internal angles (argument of the mulitplier). If you know the angle, you can compute the parameter explicitly for periods 1, 2, 3 and numerically for all periods. If I remember correctly, a simple angle is .01001000100001000001 ... times 2PI. But of course there are Siegel angles and parabolic angles which are the same for the first 100 digits.

... Maybe that is one of the reasons, why you cannot draw them..." Wolf Jung "A Cremer internal angle is obtained as 0.10001000000000000010000... with fast growing 0-gaps, but any finite approximation is parabolic." ( Wolf Jung )

External rays[edit]

Parameter ray[edit]

" For Cremer parameters, there is a unique parameter ray landing, but the dynamics is more complicated." ( Wolf Jung )


References[edit]

  1. Describing quadratic Cremer point polynomials by parabolic perturbations by DAN ERIK KRARUP SØRENSEN. Ergodic Theory and Dynamical Systems (1998), 18 : pp 739-758. 1998 Cambridge University
  2. Hedgehogs of Hausdorff dimension one by Kingshook Biswas
  3. wikipedia : Locally connected space
  4. Smooth Combs Inside Hedgehogs by Kingshook Biswas