Fractals/Iterations in the complex plane/cremer



The problem is that we are exploring environments based upon irrational numbers through computer machinery which works with finite rationals ! ( Alessandro Rosa )


 * there are Cremer Julia set for quadratic polynomial,
 * there are no images of such Julia sets
 * A Cremer Julia set has no interior and can never disconnect the plane
 * It contains single or double comb. "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
 * Cremer Julia sets are not locally connected
 * "Hedgehogs associated to an indifferent irrational non-linearizable fixed point are full non-trivial compact connected sets totally invariant by the dynamics." R. PEREZ-MARCO
 * a Cremer fixed point belongs to the Julia set

=Parameter=

For Cremer Julia set internal angle should be non-Brjuno number (irrational number, non-diophantine ).

"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. numerically there is no difference between Cremer,  locally connected Siegel,  non-locally connected Siegel,  and parabolic with a high rotation number. ... 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 )

=Model=

One can think about Cremer Julia set as Julia set with Siegel disk with infinite many digits.

(parabolic Julia set with infinite period = Parabolic filled Julia sets with a very high period will probably look similar to Siegel Julia sets )

Models
 * Rempe’s straight brush model
 * Pseudo hedgehogs by Cheritat

Hegdehogs
description
 * non-linearizable
 * Hedgehog space

ricardo perez-marco: hedgehog ressources

small cycles
"the “small cycles property”: Every neighborhood of the origin contains infinitely many periodic orbits " ( CREMER FIXED POINTS AND SMALL CYCLES by LIA PETRACOVICI )

=External rays =

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

What are the angles of external rays that land on the Cremer and Siegel parameter points c ?

Those that do not belong to any closed wake. They are irrational and I guess there is not much more known. You can approximate them with angles of suitable wakes ...

This something analogous to the construction of the middle-1/3 standard Cantor set by removing open intervals successively.
 * Start with [0,1].
 * Remove (1/3, 2/3).
 * Remove (1/7, 2/7) and (5/7, 6/7)

There is a Cantor set of angles remaining, which are the angles of all rays landing at the main cardioid. The rational angles belong to roots and the irrational angles to Siegel and Cremer parameters. Moreover, each rational angle is a boundary point of an interval removed after finitely many steps. So in the following construction of removing closed intervals, you do not get a Cantor set,  and only the irrational angles remain:
 * Start with [0,1].
 * Remove [1/3, 2/3].
 * Remove [1/7, 2/7] and [5/7, 6/7]

=Non-computable Julia sets=
 * construction

=References=
 * Pérez-Marco, R. (1994). Topology of Julia sets and hedgehogs. Orsay, France: Université de Paris-Sud, Département de mathématiques. Edition/Format: Print book : English, series Title: Prépublications (Université Paris-Sud. Département de mathématiques), no 94-48.
 * The Julia sets of quadratic Cremer polynomials. Author : Alexander Blokh, Lex Oversteegen. doi:10.1016/j.topol.2006.02.001Topology and its Applications. Volume 153, Issue 15, 1 September 2006, Pages 3038-3050
 * wikipedia : Liouville_number
 * Why golden mean is optimal: an introduction to Diophantine, Brjuno and Liouville number — Zhenghe
 * scholarpedia article by Xavier_Buff : Siegel_disks
 * open problems in hedgehogs dynamics
 * NON-ACCESSIBLE CRITICAL POINTS OF CERTAIN RATIONAL FUNCTIONS WITH CREMER POINTS by Lia Petracovici
 * Non-accessible Critical Points of Cremer Polynomials by Jan Kiwi
 * The Julia Sets of Basic Unicremer Polynomials of Arbitrary Degree Alexander Blokh
 * hThe solar Julia sets of basic quadratic Cremer polynomials by A. Blokh, X. Buff, A.Chéritat, L. Oversteege ( [Submitted on 5 Dec 2008 ]
 * hedgehog ressources