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When a complex quadratic polynomial is iterated, all points
The Mandelbrot
set is connected. Its research was pioneered
in the 1980s by A. Douady and J.H. Hubbard. They showed that

In 1996, I started to program the Mandelbrot set

In the image, strips are marked with external rays. Each strip contains a prominent frame of period 4, 7, ..., which look like stars close to the Misiurewicz point A new map
is defined by cutting the dynamic plane
into strips and sectors with external rays and choosing some iterate of
on each piece.
This quasiregular quadratic-like map is conjugated to a new quadratic
polynomial by the
Straightening Theorem.
In this example, and thus under
as well, since the dynamics are conjugate.
So the new parameter
is constructed from and straightened to
a new polynomial The new parameter

Denote the multiplier at the Misiurewicz point by and the diameter of the corresponding
frame are of the order ρ by
Tan Lei's
asymptotic similarity. Noting that the branch points of the frame have a
distance of the order ρ , I compared
images for different values of on the scales
ρ ,
ρ ... See the proof in
[A7], 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 [A7]. 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 [A3] 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 [A4]. I am also interested in homeomorphisms and similarity phenomena for other families of rational functions. See [A2, A7] and the presentation from the workshop on complex dynamics, Søminestationen in Holbæk 2009.

In 2011 I worked on the Stripping Algorithm, which converts kneading sequences to external angles. It is described in [C5] and in the presentation from the workshop on complex dynamics, Göttingen 2011. In 2013 I worked mainly on core entropy and biaccessibility dimension of quadratic polynomials [C1, C2, C3], in research collaboration with Henk Bruin, Dzmitry Dudko, Dierk Schleicher, Tan Lei, and Giulio Tiozzo. In 2015 I started to implement the Thurston Algorithm for quadratic matings with a path in moduli space, and since 2016 I am working on the papers [B1, B2, B3, B4, B5, B6, B7, B8, B9], in research collaboration with Arnaud Chéritat, Adam Epstein, Daniel Meyer, and Dierk Schleicher. The papers provide an initialization of the path for several applications, including anti-matings, and give results on the convergence of slow mating and equipotential gluing. There are examples of mating discontinuity, shared matings, bounded and unbounded ray connections, and a discussion of Lattès matings as well. See the videos of matings, anti-matings, and captures. Some results are announced on the poster from the conference in honor of John H. Hubbard, Bremen 2015, and on the poster from the conference celebrating John Milnor, Cancún 2016 (which I attended only virtually). Convergence results were presented at the workshop on complex dynamics, Søminestationen in Holbæk 2016, and in Toulouse, Warwick, and Bremen as well. See this presentation and a more recent one from Paris.

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.

Math 215, University of Rochester.

Mandelbrot set explorer and interactive papers by Bob Devaney.

—Hide this text ...

When a complex quadratic polynomial

*f*

_{c}(z)=z^{2}+c*z*with large modulus escape to ∞. The non-escaping points form the filled Julia set*K*.

_{c}*M*consists of those parameters*c*, such that*K*

_{c}*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 [C0]. 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 ... [C4]. These are structured by two branch points with three branches each, but they look 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 [A1], 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 contains a prominent frame of period 4, 7, ..., which look 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*g*

_{c}(z)*f*

_{c}(z)*f*

_{d}(z)=z^{2}+d*z*=0 is 4-periodic under*g*

_{c}(z)*f*

_{d}(z)*d*is the center of period 4. Now for any parameter*c*on the left edge, a new map*g*

_{c}(z)*f*

_{c}(z)*f*.

_{d}(z)*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 [A5] 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 [A2]. The homeomorphisms constructed here are approximately linear at the Misiurewicz points bounding their domains. In [A6], 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

*c*. Then_{n}*c*

_{n}-a^{-n}

^{-3/2 n}

*n*and found asymptotic models for*M-c*

_{n}^{-3/2 n}

^{-7/4 n}

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 [A7]. 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 [A3] 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 [A4]. I am also interested in homeomorphisms and similarity phenomena for other families of rational functions. See [A2, A7] and the presentation from the workshop on complex dynamics, Søminestationen in Holbæk 2009.

In 2011 I worked on the Stripping Algorithm, which converts kneading sequences to external angles. It is described in [C5] and in the presentation from the workshop on complex dynamics, Göttingen 2011. In 2013 I worked mainly on core entropy and biaccessibility dimension of quadratic polynomials [C1, C2, C3], in research collaboration with Henk Bruin, Dzmitry Dudko, Dierk Schleicher, Tan Lei, and Giulio Tiozzo. In 2015 I started to implement the Thurston Algorithm for quadratic matings with a path in moduli space, and since 2016 I am working on the papers [B1, B2, B3, B4, B5, B6, B7, B8, B9], in research collaboration with Arnaud Chéritat, Adam Epstein, Daniel Meyer, and Dierk Schleicher. The papers provide an initialization of the path for several applications, including anti-matings, and give results on the convergence of slow mating and equipotential gluing. There are examples of mating discontinuity, shared matings, bounded and unbounded ray connections, and a discussion of Lattès matings as well. See the videos of matings, anti-matings, and captures. Some results are announced on the poster from the conference in honor of John H. Hubbard, Bremen 2015, and on the poster from the conference celebrating John Milnor, Cancún 2016 (which I attended only virtually). Convergence results were presented at the workshop on complex dynamics, Søminestationen in Holbæk 2016, and in Toulouse, Warwick, and Bremen as well. See this presentation and a more recent one from Paris.

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.

Math 215, University of Rochester.

Mandelbrot set explorer and interactive papers by Bob Devaney.

—Hide this text ...

### Papers on homeomorphisms and self-similarity:

[A1]**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.

[A2]

**Homeomorphisms of the Mandelbrot Set**

arXiv:math/0312279. Appeared in

*Dynamics on the Riemann Sphere*, A Bodil Branner Festschrift,

EMS 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.

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.

[A3]

**Renormalization and embedded Julia sets in the Mandelbrot set**

Preprint in preparation (2020). Summary.

X
Review of holomorphic motions, transversal sections, simple renormalization.
Combinatorial construction and structure of embedded Julia sets.
Relation to asymptotic and local similarity.

*incomplete*pdf.

[A4]

**Primitive and satellite renormalization of quadratic polynomials**

Preprint in preparation. Summary.

X
Review of holomorphic motions, transversal sections,
straightening of quadratic-like maps.
Self-contained presentation of primitive and
satellite renormalization, including combinatorics, loci, landing properties.

[A5]

**Quasiconformal and combinatorial surgery**

Preprint in preparation. 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.

[A6]

**Self-similarity and homeomorphisms of the Mandelbrot set**

Preprint in preparation. Summary.

X
Combinatorial description of fundamental domains at Misiurewicz
points. Construction of corresponding homeomorphisms.
Review of asymptotic self-similarity.

[A7]

**Local and asymptotic similarity of the Mandelbrot set and Julia sets**

Preprint in preparation. Summary.

X
Local similarity between the decorations of small Mandelbrot sets and Julia
sets. Comparison to asymptotic similarity on multiple scales.
Non-hairiness of decorations. Generalization to other parameter spaces.

### Papers on the Thurston Algorithm and matings:

[B1]**Quadratic matings and anti-matings**

arXiv:2012.. Preprint of December 2020. Summary.

X
Anti-mating means that two planes or half-spheres are mapped to each
other by quadratic polynomials, and the filled Julia sets of two quartic
polynomials are glued together. The slow Thurston Algorithm is implemented
and shown to converge. An existence criterion is given, which is analogous
to the non-conjugate limbs condition for matings, and the notion of an
anti-equator is discussed. For specific families of quadratic rational maps,
the loci of mating, anti-mating, and captures are obtained conjecturally.

*incomplete*pdf.

[B2]

**Topological matings and ray connections**

arXiv:1707.00630. Preprint of July 2017, update in preparation.

Summary.

X
For formal matings of certain classes of geometrically finite and infinite
polynomials, the shape and diameter of ray-equivalence classes
is described explicitly, and the topological mating can be constructed
without employing the Rees--Shishikura--Tan Theorem to exclude cyclic ray
connections. On the other hand, unbounded cyclic ray connections are
found when P is primitive renormalizable and Q is chosen appropriately;
then the topological mating is not even defined on a pinched sphere, but
there is no Hausdorff space at all. In addition, matings with long ray
connections are found alogithmically.

*preliminary*pdf. Related software.

[B3]

**Convergence of the Thurston Algorithm for quadratic matings**

arXiv:1706.04177. Preprint of June 2017, update in preparation.

Summary.

X
When the Thurston Algorithm for the formal mating diverges in ordinary
Teichmueller space due to postcritical
identifications, it still converges on the level of rational maps and colliding
marked points --- it is not necessary to implement the essential mating by
encoding ray-equivalence classes numerically. The proof is based on the
extended pullback map on augmented Teichmueller space constructed by Selinger.

*preliminary*pdf.

[B4] Jointly with Arnaud Chéritat:

**Slow mating and equipotential gluing**

Preprint in preparation, based in part on Section 5 in arXiv:1706.04177v1 of 2017.

Summary.

X
Equipotential gluing is an alternative definition of mating, which is not based
on the Thurston Algorithm. Equipotential lines of the two polynomials are glued
to define maps between spheres, and the limit of potential 0 is considered.
The initialization of the slow mating algorithm depends on an initial radius
R; when R goes to infinity, slow mating is shown to approximate equipotential
gluing. The visualization in terms of holomorphically moving Julia sets and
their convergence is discussed as well, and in the periodic case a conformal
mating in the strongest sense is obtained: the semi-conjugation is the limit
of a holomorphic motion.

[B5]

**Lattès maps and quadratic matings**

Preprint in preparation. Summary.

X
Lattes maps of type (2, 2, 2, 2) or (2, 4, 4) are represented by matings in
basically nine, respectively three, different ways. The proof combines
combinatorics of polynomials and ray-equivalence classes with the Shishikura
Algorithm relating the topology of the formal mating to the multiplier of the
corresponding affine map on a torus. This shows that all matings from
non-conjugate limbs exist, which does not follow from the well-known
absence of obstructions. The slow mating algorithm diverges in certain cases:
while the expected collisions are happening, a neutral eigenvalue
from the one-dimensional Thurston Algorithm persists, producing an attracting
center manifold in configuration space. (Joint work with Arnaud Cheritat.)

*incomplete*pdf.

[B6]

**Hurwitz equivalence and Lattès maps**

Preprint in preparation. Summary.

X
The Hurwitz equivalence between quadratic rational maps with the same
ramification portrait is constructed explicitly, complementing the approach
related to the moduli space map by Sarah Koch. Twisted Lattes maps and the
virtual endomorphism of Lattes maps are discussed as well.

*preliminary*pdf.

[B7]

**Quadratic captures and matings**

Preprint in preparation, based in part on Sections 2.3 and 6 in arXiv:1706.04177v1 of 2017.

Summary.

X
An alternative construction of quadratic matings by encapture, a
repelling-preperiodic capture is obtained. The Thurston Algorithm is shown to
converge with identifications, and its implementation is discussed as well.

*incomplete*pdf.

[B8]

**Ray connections and shared matings**

Preprint in preparation, based on Section 3 in arXiv:1707.00630v1 of 2017.

Summary.

X
Based on combinatorics of ray connections, simple examples of mating
discontinuity and of unboundedly shared matings are given. Here the
multiplicity grows linearly with preperiod and period. In some cases, upper
bounds on the multiplicity are obtained as well.

*preliminary*pdf.

[B9]

**The Thurston Algorithm for quadratic polynomials**

Preprint in preparation. Summary.

X
The slow Thurston algorithm is implemented for several kinds of Thurston maps
giving quadratic polynomials. These include a spider algorithm with a path
instead of legs, Dehn twisted polynomials, moving the critical value by
recapture or encapture, and tuning. Using the Selinger results on removable
obstructions, the spider algorithm is shown to converge in the obstructed
case of satellite Misiurewicz points as well. Recapture surgery is related
to internal addresses, and used to discuss a specific example of twisted
polynomials.

### Papers on core entropy, combinatorics, and external rays:

[C0]**Some Explicit Formulas for the Iteration of Rational Functions**

Unpublished manuscript of 1997. Download pdf.

[C1,C2]

**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.

*Erratum: Lemma 4.1 needs to be modified in the Siegel case.*

[C3]

**Core entropy and biaccessibility dimension**, Appendix A in:

D. Dudko, D. Schleicher, Core entropy of quadratic polynomials.

arXiv:1412.8760. Arnold Mathematical Journal (2020),

https://doi.org/10.1007/s40598-020-00134-y.

[C4]

**Edges and frames in the Mandelbrot set**

Preprint in preparation. 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.

[C5]

**Combinatorics and external rays of the Mandelbrot set**

Preprint in preparation. Summary.

X
Review of combinatorial descriptions by external angles, Hubbard trees,
kneading sequences, and internal addresses.
Discussion and proof of the Stripping Algorithm, which finds
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.

### Papers on scattering theory and quantum mechanics:

[E0]**Multiple Reflections in One-Dimensional Quantum Scattering**

Unpublished manuscript of 1998. Download pdf.

[E1]

**Der geometrische Ansatz zur inversen Streutheorie bei der Dirac-Gleichung**

Diploma thesis of 1996. Download pdf.

[E2]

**Geometrical Approach to Inverse Scattering for the Dirac Equation**

Appeared in Journ. Math. Phys., vol

**38**, January 1997, pp 39 - 48.

The original article is found there. Copyright 1997 American Institute of Physics.

Download pdf here. This article may be downloaded for personal use only.

[E3] Jointly with V. Enss:

**Geometrical Approach to Inverse Scattering**

Appeared in the proceedings of the First MaPhySto Workshop on Inverse Problems,

April 1999, Aarhus. MaPhySto Miscellanea no. 13, 1999, ISSN 1398-5957. Download pdf.

[E4]

**Gauge Transformations and Inverse Quantum Scattering with Medium-Range Magnetic Fields**

arXiv:math-ph/0412096. Appeared in Mathematical Physics Electronic Journal MPEJ,

vol

**11**, paper 5, December 2005, 32 pp. Freely available at MPEJ. Download pdf here.

[E5]

**Inverse Relativistic and Obstacle Scattering with Medium-Range Magnetic Fields**

Preprint in preparation. A 2-page summary was added to the previous preprint.

### Papers on fracture mechanics and composite materials:

[F1] Jointly with B. Banholzer, W. Brameshuber, J. Geus:**Bestimmung eines Verbundgesetzes auf Basis von Einzelfaser-Pull-Out-Versuchen**

Appeared in Bautechnik vol

**81**, October 2004, pp 806 - 812. The original article is found there.

[F2] Jointly with B. Banholzer, W. Brameshuber:

**Analytical simulation of pull-out tests — the direct problem**

Appeared in Cement and Concrete Composites vol

**27**, January 2005, pp 93 - 101.

The original article is found there.

[F3] Jointly with B. Banholzer, W. Brameshuber:

**Analytical evaluation of pull-out tests — The inverse problem**

Appeared in Cement and Concrete Composites vol

**28**, July 2006, pp 564 - 571.

The original article is found there.