Fractals/Iterations in the complex plane/cremer
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< Fractals
- 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]
- it is hard to give c -values
"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
References [edit]
- ↑ 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
- ↑ Hedgehogs of Hausdorff dimension one by Kingshook Biswas
- ↑ wikipedia : Locally connected space
