Home Scattering Mandelbrot Fracture Software Mandel Software Binomial

• Overview • Introduction • Asymptotic scaling • Local similarity • Embedded Julia sets

Wolf Jung   jung@mndynamics.com • Deutsch
Gesamtschule Aachen-Brand, Rombachstrasse 99, 52078 Aachen, Germany.


An interactive exposition of complex dynamics is contained as a demo in the program Mandel.
A short introduction is given here with the Java applet Juliette. Online courses at other sites:
Mandelbrot set anatomy by Evgeny Demidov.
Mu-Ency glossary and encyclopedia by Robert Munafo.
Mandelbrot set at Wikipedia.
Logistic equation by Arnaud Chéritat.
Order course at the M. Casco Learning Center.
Fractal geometry and the Mandelbrot set at Yale University.
Math 215, University of Rochester.
Mandelbrot set explorer and interactive papers by Bob Devaney.

Research on the Mandelbrot set:

When a complex quadratic polynomial fc(z)=z2+c is iterated, all points z with large modulus are escaping to ∞. The non-escaping points form the filled Julia set Kc. The Mandelbrot set M consists of those parameters c, such that Kc is connected. Its research was pioneered in the 1980s by A. Douady and J.H. Hubbard. They showed that M is connected, described it combinatorially by the landing properties of external rays, and explained the appearance of small Mandelbrot sets within M by renormalization. The mathematical field of complex dynamics became active by the use of quasiconformal maps and by the inspiration from computer graphics. The fascination was shared by physicists, programmers, artists ...

In 1996, I started to program the Mandelbrot set M in Qbasic, then C++, and later in Java. I was interested in drawing external rays and in explicit formulas for periodic points. The latter had been found by E. Netto in 1900 and were rediscovered a few times. See the references in [10]. Soon I was fascinated by the principal branch point a of the 1/3-limb of M, a Misiurewicz point with preperiod 3 and period 1. The left and right branches can be described as a union of “frames”: on the left edge there is one of period 4, two of period 7, four of period 10 ... [7]. These are structured by two branch points with three branches each, but they are looking like stars with six branches when they are close to a. In 1999 I noted that the techniques of quasiconformal surgery by Branner-Douady and Branner-Fagella could be adapted to construct a homeomorphism between frames. This formed the basis of my thesis [1], written under the direction of V. Enss, G. Jank and W. Bergweiler, in research collaboration with J. Riedl and D. Schleicher.

In the image, strips are marked with external rays. Each strip is containing a prominent frame of period 4, 7, ..., which are looking like stars close to the Misiurewicz point a at the center. The homeomorphism h in the parameter plane is expanding at a, mapping each strip to the next one. Consider a center parameter c of period 7: the critical point z=0 is 7-periodic under the iteration of the quadratic polynomial fc(z)=z2+c. A new map gc(z) is defined by cutting the dynamic plane into strips and sectors with external rays and choosing some iterate of fc(z) on each piece. This quasiregular quadratic-like map is conjugated to a new quadratic polynomial fd(z)=z2+d by the Straightening Theorem. In this example, z=0 is 4-periodic under gc(z) and thus under fd(z) as well, since the dynamics are conjugate. So the new parameter d is the center of period 4. Now for any parameter c on the left edge, a new map gc(z) is constructed from fc(z) and straightened to a new polynomial fd(z). The new parameter d depends on the old parameter c, and the map in parameter space is defined by h(c)=d. It is shown to be continuous and bijective, i.e., a homeomorphism. See [4] for details, and demo 8 of Mandel. This construction is possible under general combinatorial assumptions. Results on the homeomorphism group of M are obtained in [2]. The homeomorphisms constructed here are approximately linear at the Misiurewicz points bounding their domains. In [5], homeomorphisms are constructed at given Misiurewicz points. See the presentation from the conference on complex analysis, Bedlewo 2014.

Denote the multiplier at the Misiurewicz point a by ρ and the center of period n+7 by cn. Then cn-a and the diameter of the corresponding frame are of the order ρ-n by Tan Lei's asymptotic similarity. Noting that the branch points of the frame have a distance of the order ρ-3/2 n, I compared images for different values of n and found asymptotic models for M-cn on the scales ρ-3/2 n, ρ-7/4 n... See the proof in [5], demo 6 of Mandel, and the interactive exploration with a Java applet here.

A local similarity between the decorations of small Mandelbrot sets and small Julia sets has been observed by H.-O. Peitgen. See his 1988 book. In 2005 I came to believe that this phenomenon was related to an approximately affine renormalization, which I checked with the program, and later found a proof [3]. See also demo 7 of Mandel, the interactive exploration with a Java applet here, and the presentation from the workshop on complex dynamics, Søminestationen in Holbæk 2007.

Recent results on embedded Julia sets [3] are illustrated in demo 5 of Mandel and here with a Java applet. I am also interested in homeomorphisms and similarity phenomena for other families of rational functions. See [2,5,6] and the presentation from the workshop on complex dynamics, Søminestationen in Holbæk 2009.

The Stripping Algorithm is converting kneading sequences to external angles. It is described in [9] and in the presentation from the workshop on complex dynamics, Göttingen 2011. The phenomenon of early returns led to the discussion of Dehn twists and the definition of recapture surgery based on the Thurston algorithm [10]. In 2013 I worked mainly on core entropy and biaccessibility dimension [8].

Papers on homeomorphisms and self-similarity:

[1] Homeomorphisms on Edges of the Mandelbrot Set
Ph.D. thesis of 2002. Available from the RWTH library, the IMS thesis server,
and here as pdf (1.8 Mb).

[2] Homeomorphisms of the Mandelbrot Set
arXiv:math/0312279. Appeared in Dynamics on the Riemann Sphere,
A Bodil Branner Festschrift, Poul G. Hjorth and Carsten Lunde Petersen Editors, EMS,
January 2006, pp 139-159. ISBN 3-03719-011-6. Summary. Download pdf.

[3] Renormalization, local similarity, and embedded Julia sets in the Mandelbrot set
arXiv:1408.. Preprint of August 2014. Summary. Download pdf.

[4] Quasiconformal and combinatorial surgery
Preprint in preparation (December 2014). Summary. Download pdf.

[5] Self-similarity and homeomorphisms of the Mandelbrot set
Preprint in preparation (September 2014). Summary. Download pdf.

[6] Surgery for Newton-like maps
Preprint in preparation (2016). Summary. Download pdf.

Papers on combinatorics and external rays:

[7] Edges and frames in the Mandelbrot set
Preprint in preparation (2015). Summary.

[8] Core entropy and biaccessibility of quadratic polynomials
arXiv:1401.4792. Preprint of January 2014. Summary. Download pdf.

[9] Combinatorics and external rays of the Mandelbrot set
Preprint in preparation (2015). Summary. Download pdf.

[10] Applications and implementation of the Thurston Algorithm for quadratic maps
Preprint in preparation (2015). Summary. Download pdf.
The appendix contains a revised version of:
Some Explicit Formulas for the Iteration of Rational Functions
Unpublished manuscript of 1997. Download pdf.


Last modified: July 2014.  Disclaimer: I am not responsible for linked sites by other people.
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