# Seminar: Dynamics in one complex variable

Sommersemester 2021

## Jun-Prof. Dr. Maria Beatrice Pozzetti

## Jun-Prof. Dr. Gabriele Benedetti

### Schedule

- As a block seminar on Saturday June 12 and Saturday June 19 Online on zoom

### Organization Meeting

- Thursday March 11, 16:15-18:00 Uhr, on zoom

There are still talks available! If you are interested in the seminar please send us an e-mail (gbenedetti@mathi.uni-heidelberg.de and pozzetti@mathi.uni-heidelberg.de)

### Müsli and MaMpf

Please, register for the seminar on MÜSLI and on MaMpf

### Description

In this seminar, we study both theoretically and experimentally the dynamics of holomorphic mappings f on a Riemann surface S. Key examples will be rational functions on the Riemann sphere as simple as f(z)=z^2+2. The crucial question in this framework is to understand the behaviour of the sequence z, f(z), f(f(z)),... with starting point in S. Is the sequence periodic? Is it converging? How does its behaviour change if we move the starting point? The starting points giving rise to a chaotic behaviour under the above procedure form the so-called Julia set of the mapping f. More generally, we will consider a family f_t of functions parametrized by a complex number t. The values of t for which the Julia set is connected will form the so-called Mandelbrot set in parameter space. The Julia and Mandelbrot sets have a beautiful fractal structure and complicated topological properties. We aim to understand them rigorously and visualize them experimentally with a computer during the seminar.

At the organization meeting several topics will be proposed. Some will focus more on the mathematical foundations, other on visualisation and experimentation. Among them, you can choose one topic you are interested in. Once you have agreed on the topic of your project with us, you should contact us to discuss the content of your talk in more detail. Please let us help you during the preparation of your presentation.

Here is a syllabus of the talks.

### Prerequisites

This seminar is aimed at students who are interested in geometry. Familiarity with the basics courses in a bachelor curriculum in math (Linear algebra, Analisis, Funktionentheorie) is expected. There are several topics for students to choose from ranging in difficulty and prerequisites. The seminar is open for bachelor and master students, the talks will be given in English or German, depending on each speaker's preference.

### Schedule of the talks

11.03.2020 | Organization meeting | ||

10.06.2020 | Teaser | ||

16:00 | Marcel | V1: Julia-Fatou | |

12.06.2020 | First day | ||

9:30 | Monika | S2: Pick | |

11 | Burcu | S.3: Montel | |

14:00 | Anna | V2: Mandelbrot | |

19.06.2020 | Second day | ||

9:30 | Franziska | S.4: Julia and Fatou | |

10:40 | Cornelius | S.7: Indifferent points | |

11:50 | Britta | S.8: Attracting points | |

14:00 | Julia | S.10:Parabolic | |

15:10 | Eric | S.11:Cremer |

### References

- John Milnor, Dynamics in one complex variable. Third edition. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, viii+304 pp., 2006.
- Robert L. Devaney, A first course in chaotic dynamical systems: theory and experiment. Second edition. Boca Raton: CRC Press, Taylor & Francis Group, x+318 pp., 2020.
- Curtis McMullen, Complex Dynamics and Renormalization. Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, x+214 pp., 1994.
- Alan F. Beardon, Iteration of rational functions, Complex analytic dynamical systems. Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991. xvi+280 pp.
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics. Universitext: Tracts in Mathemat- ics. Springer-Verlag, New York, x+175 pp., 1993.