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.
X Sketch of a general result on quasiconformal surgery, which turns combinatorial data into homeomorphisms. Examples.
Definition of non-trivial homeomorphism groups. These are totally disconnected and they have the cardinality of R. Generalization to other parameter spaces, e.g., cubic Newton maps.
Download pdf.

[3] Renormalization, local similarity, and embedded Julia sets in the Mandelbrot set
Preprint in preparation (2015). Summary.
X Review of holomorphic motions, transversal sections, straightening of quadratic-like maps. Self-contained presentation of primitive and satellite renormalization.
Local similarity between the decorations of small Mandelbrot sets and Julia sets. Construction of embedded Julia sets.
Download pdf.

[4] Quasiconformal and combinatorial surgery
Preprint in preparation (2016). Summary.
X Straightening of quasiregular quadratic-like maps. General construction of homeomorphisms of the Mandelbrot set from combinatorial assumptions. Description and alternative construction by mapping external angles. Examples of homeomorphisms on generalized edges.
Download pdf.

[5] Self-similarity and homeomorphisms of the Mandelbrot set
Preprint in preparation (2016). Summary.
X Combinatorial description of fundamental domains at Misiurewicz points. Construction of corresponding homeomorphisms. Review of asymptotic self-similarity. Generalization to multiple scales around small Mandelbrot sets. Local similarity, asymptotics, and non-hairiness of decorations. Generalization to other parameter spaces.
Download pdf.

[6] Surgery for Newton-like maps
Preprint in preparation (2017). Summary. Download pdf.

Papers on combinatorics, external rays, and rational maps:

[7] Edges and frames in the Mandelbrot set
Preprint in preparation (2016). Summary.
X Correspondence between puzzle pieces and para-puzzle pieces. Stepwise construction of new para-puzzle pieces corresponding to preimages of a puzzle piece that corresponds to a known para-puzzle piece. Examples: limbs, edges, and frames.
Download pdf.

[8] Core entropy and biaccessibility of quadratic polynomials
arXiv:1401.4792. Preprint of January 2014. Summary.
X Markov matrices for postcritically finite Hubbard trees are combined with continuity results to discuss core entropy and biaccessibility dimension of quadratic polynomials. Specifically, results on monotonicity, level sets, renormalization, Hoelder asymptotics and self-similarity are obtained.
Download pdf.
See also the appendix of arXiv:1412.8760.

[9] Combinatorics and external rays of the Mandelbrot set
Preprint in preparation (2015). Summary.
X Review of combinatorial descriptions by external angles, Hubbard trees, kneading sequences, and internal addresses. Discussion and proof of the Stripping Algorithm, which is finding external angles by iterating strips or intervals backwards according to a given kneading sequence. Early returns to the characteristic wake. Implementation details. Review of Thurston Algorithm, Spider Algorithm, twisted rabbits. Recapture surgery. Example with Dehn twists and early returns.'
Download pdf.

[10] Applications and implementation of the Thurston Algorithm for quadratic maps
Preprint in preparation (2015). Summary.
X Review of Dehn twists, Thurston Algorithm, Spider Algorithm. Implementation with a path in moduli space. Application to polynomials, matings, anti-matings, and other kinds of surgery. Recapture surgery. Example with Dehn twists and early returns.
Download pdf.

[0] Some Explicit Formulas for the Iteration of Rational Functions
Unpublished manuscript of 1997. Download pdf.

Research on the Mandelbrot set and complex dynamics:

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 Netto in 1900 and were rediscovered a few times. See the references in [0]. 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.

More recent results on embedded
Julia sets [3] are illustrated in demo 5 of Mandel and here with a Java applet. The research on local similarity and embedded Julia sets was accompanied by working out folk results on renormalization. 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.

In 2013, I worked mainly on core entropy and biaccessibility dimension of quadratic polynomials [8], in research collaboration with Henk Bruin, Dzmitry Dudko, Dierk Schleicher, Tan Lei, and Giulio Tiozzo. 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 2015, I am trying to finish some old papers, especially [3] of 2010–2012–2014 and [9] of 2011, and implementing the Thurston algorithm with a path in moduli space. See these videos of matings, anti-matings, and captures. And see the poster from the conference in honor of John H. Hubbard, Bremen 2015.

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.