## 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,

139-159, EMS 2006. 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.

[A3]

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

Preprint in preparation (2024). Summary.

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

*preliminary*pdf.

[A4]

**Primitive and satellite renormalization of quadratic polynomials**

Preprint in preparation (2025). Summary.

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

[A5]

**Quasiconformal and combinatorial surgery**

Preprint in preparation (2025). Summary.

X
Straightening of quasi-regular 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 (2025). 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 (2025). 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]**Topological matings and ray connections**

arXiv:1707.00630. Preprint of 2017, update in preparation (2024).

Summary.

X
For formal matings of certain classes of geometrically finite and
infinite polynomials, the structure 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. Moreover, matings with long ray
connections are found alogithmically.

*preliminary*pdf. Related software.

[B2]

**Ray connections and shared matings**

Preprint in preparation (2024), 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.

[B3]

**Lattès maps and quadratic matings**

arXiv:2406.. Preprint of June 2024. Summary.

X
Lattès 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, which relates 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, since the orbifold (2, 2, 2, 2) is not hyperbolic.

*preliminary*pdf.

[B4]

**Hurwitz equivalence and quadratic Lattès maps**

arXiv:2406.. Preprint ofJune 2024. 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 Lattès maps, the
pullback relation of curves, and the virtual endomorphism of Lattès maps are
discussed, using the lift to a real affine map. Questions of equivalence are
related to reduction of quadratic forms (joint work with Michael H. Mertens).

*incomplete*pdf.

[B5]

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

arXiv:1706.04177. Preprint of 2017, update in preparation (July 2024).

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. An
informal introduction to his results is included, and a strategy for proving
the Thurston Theorems from the Canonical Obstruction Theorem is outlined.

*preliminary*pdf.

[B6] Jointly with Arnaud Chéritat:

**Convergence of slow mating**

Preprint in preparation (June 2024), 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. On the other hand, for matings of Lattès type (2, 2, 2, 2)
the slow mating algorithm diverges in general: while the expected collisions
are happening, a neutral eigenvalue from the one-dimensional Thurston Algorithm
persists, producing an attracting center manifold in configuration space.

[B7]

**Quadratic polynomials, captures, and matings**

Preprint in preparation (2024), based in part on Sections A, 2.3, and 6 in arXiv:1706.04177v1 of 2017.

Summary.

X
Various Thurston maps are defined by moving a critical value along a path, which
is simple to implement numerically and to visualize by progressive
identifications. This includes a spider algorithm with a path instead of legs,
Dehn twisted polynomials, moving the critical value by capture or recapture. The
spider algorithm is shown to converge in the obstructed case of satellite
Misiurewicz points as well. An alternative construction of quadratic matings by
a repelling capture is due to Mary Rees, which can be used to obtain shared
matings. Moreover, regluing at capture components and examples from V3 are
discussed, including one-sided components and indirect captures.

*incomplete*pdf. See also the videos.

[B8]

**Quadratic matings and anti-matings**

Preprint in preparation (July 2024). 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. An existence criterion due to Ahmadi Dastjerdi is discussed,
which is analogous to the non-conjugate limbs condition for matings.
An anti-equator is sufficient, and in the hyperbolic case necessary, for an
anti-mating. The loci of mating and anti-mating are obtained conjecturally
for specific families of quadratic rational maps.

and this presentation. Download

*incomplete*pdf.

## 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,
Hölder asymptotics and self-similarity are obtained.

—Meanwhile, continuity has been shown by Tiozzo and Dudko–Schleicher, as well as maximality.

—Meanwhile, continuity has been shown by Tiozzo and Dudko–Schleicher, as well as maximality.

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

[C3]

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

Dzmitry Dudko, Dierk Schleicher, Core entropy of quadratic polynomials.

arXiv:1412.8760. Arnold Mathematical Journal

**6**, 333-385 (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, and implementation details.

## 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 (in German). 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 Volker 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.

## Papers on Processwork:

[G0]**Physik und Prozessarbeit**

Manuscript of a presentation at Institut für Prozessarbeit Deutschland in June 2020 (in German).

A revised version is in preparation. Summary.

X
Processwork (or Process Oriented Psychology) is a method of awareness training
founded by Arnold Mindell in the 1980s. It is applied in various areas,
including personal therapy and facilitation. A basic idea is to help the client
to become aware of secondary processes and to integrate them. For example, they
may identify as being peaceful and be disturbed by violent fantasies; after
exploring both violent inner figures and the opposing belief system, the client
shall be able to express their wishes and needs, as well as their boundaries,
more directly. Alternatively, or in addition, the therapist may help the client
to access the quality of being more direct as an essence of the violent
fantasies, without exploring and amplifying the latter explicitly.
Arnold Mindell developed and explained essence work from his background as
a physicist, as well as his experience with Jungian psychology, Daoism and
shamanism. Here I aim to present ideas from both quantum physics and
Processwork to a general audience, giving examples of concepts seen as
analogous. And I emphasize that in my opinion, these connections between
physics and psychology need to be framed clearly as analogies: there is no
claim whatsoever, that quantum phenomena guide cognitive processes in the brain.

[G1]

**Trauma, Empowerment, and Compassion**

Project paper for completing the basic training at Institut für Prozessarbeit Deutschland in July 2021.

Summary.

X
Processwork (or Process Oriented Psychology) is a method of awareness training
with various areas of application, including psychotherapy and facilitation.
Here tools and attitudes from Processwork are illustrated with my personal
process of dealing with a traumatic experience: On July 29, 2019, a boy was
shoved onto the tracks in Frankfurt main station, and killed by the incoming
train. Witnessing this, and hearing the mother cry out in agony, I felt
horrified like never before in my life. At the same time, I felt determined to
work through it as deeply and consciously as possible; I came to find a deeper
sense of love and compassion for my fellow human beings, as well as
self-compassion and empowerment, and I am still learning to speak about this
process.