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• 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.

When a complex quadratic polynomial
*f _{c}(z)=z^{2}+c*

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
*f _{c}(z)=z^{2}+c*.

Denote the multiplier at the Misiurewicz point *a* by ρ and the
center of period *n*+7*c _{n}*. Then

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].

[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:1502..
Preprint of February 2015. Summary. Download pdf.

[4]
**Quasiconformal and combinatorial surgery**

Preprint in preparation (2016). Summary. Download pdf.

[5]
**Self-similarity and homeomorphisms of the Mandelbrot set**

Preprint in preparation (2016). Summary. Download pdf.

[6]
**Surgery for Newton-like maps**

Preprint in preparation (2017). Summary. Download pdf.

[7]
**Edges and frames in the Mandelbrot set**

Preprint in preparation (2016). Summary.

[8]
**Core entropy and biaccessibility of quadratic polynomials**

arXiv:1401.4792.
Preprint of January 2014. Summary. Download pdf.

See also the appendix of arXiv:1412.8760.

[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: January 2015. Disclaimer: I am not responsible for linked sites by other people.

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