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.
Download pdf.

[A3] Renormalization and embedded Julia sets in the Mandelbrot set
Preprint in preparation (2021). Summary.
X Review of holomorphic motions, transversal sections, simple renormalization. Combinatorial construction and structure of embedded Julia sets. Relation to asymptotic and local similarity.
Download 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.
Download pdf.

[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.
See [A2].

[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.
See this presentation.

[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.
See this presentation, and generalizations here.

Papers on the Thurston Algorithm and matings:

[B1] Quadratic matings and anti-matings
arXiv:2107.. Preprint of July 2021. 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.
See also the videos. Download incomplete pdf.

[B2] Topological matings and ray connections
arXiv:1707.00630. Preprint of July 2017, update in preparation.
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.
Download preliminary pdf. Related software.

[B3] Convergence of the Thurston Algorithm for quadratic matings
arXiv:1706.04177. Preprint of June 2017, update in preparation.
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.
Download 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.
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.
See this presentation. Related software.

[B5] Lattès maps and quadratic matings
Preprint in preparation. 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.
Download 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 Lattès maps, the pullback relation of curves, and the virtual endomorphism of Lattès maps are discussed as well.
Download 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.
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. In the postcritcally finite case, regluing at capture components and examples are discussed from this perspective.
Download incomplete pdf.

[B8] Ray connections and shared matings
Preprint in preparation, based on Section 3 in arXiv:1707.00630v1 of 2017.
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.
Download 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.
See the appendix of arXiv:1706.04177v1.

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.
Download pdf.
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),

[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.
See [A1].

[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.
See this presentation.

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 and will be available within a few weeks. 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.
Download pdf.

[G1] Trauma, Empowerment, and Compassion
Project paper for completing the basic training at Institut für Prozessarbeit Deutschland in July 2021.
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.
Download pdf.