# Sébastien Godillon

### Personal data:

I am Doctor of Mathematics. You may contact me at the following address:

 E-mail: seb "dot" godillon "at" gmail "dot" com

Previously, I was Ph.D. student and teaching assistant from September 2005 to August 2008 and then teaching and research assistant at the Department of Mathematics of the University of Cergy-Pontoise. From January to September 2009, I was a Marie Curie early stage research fellow at the Department of Mathematics and Physics of the University of Roskilde within the European research training network CODY. I held also postdoc positions from September to December 2010 and from September to December 2011 at the Mathematical institute of the Chinese Academy of Sciences in Beijing, and from September 2012 to December 2013 at the Department of Mathematics of the University of Barcelona. My detailed curriculum vitae:

### Research activities:

I am studying holomorphic dynamical systems and more precisely those ones coming from the iteration of rational maps on the Riemann sphere. The theory of holomorphic dynamical systems has many links with several fields of mathematics. So, I am also interested in complex analysis, discrete dynamical systems in general and symbolic dynamics in particular, complex analytic geometry (Teichmüller theory), and algebraic topology (classification of ramified coverings and iterated monodromy groups).

I have obtained my Ph.D. at the University of Cergy-Pontoise, under the supervision of Tan Lei. The title of my thesis is "Construction of rational maps with prescribed dynamics". The subject is the realisation problems (existence and effective construction) of some combinatorial dynamics by rational maps. The tools used are dynamics on topological trees, quasiconformal surgery and a powerful Thurston's theorem which provides a combinatorial characterization of rational maps in the family of post-critically finite ramified coverings. I defended my thesis on May 12, 2010.

Here is a list of articles (an abstract is available for each):

• Published:
• Introduction to Iterated Monodromy Groups (pdf) (arxiv)
(abstract)

#### Introduction to Iterated Monodromy Groups

The theory of iterated monodromy groups was developed by Nekrashevych. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych.

These notes attempt to introduce this theory for those who are familiar with holomorphic dynamics but not with group theory. The aims are to give all explanations needed to understand the main definition and to provide skills in computing any iterated monodromy group efficiently. Moreover some explicit links between iterated monodromy groups and holomorphic dynamics are detailed. In particular, a classification of quadratic branched coverings with three post-critical points is provided in terms of iterated monodromy group, and matings of polynomials are discussed.

Annales de la facultés des sciences de Toulouse, sér. 6, 21 n°S5 (2012), p. 1069-1118
• Accepted for publication:
• A family of rational maps with buried Julia components (pdf) (arxiv)
(abstract)

#### A family of rational maps with buried Julia components

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least 5.

This paper introduce a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of critically separating Julia components is encoded by a weighted Hubbard tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover this family contains some rational maps of degree 3 with explicit formula that answers a question McMullen raised.

to appear in Ergodic Theory and Dynamical Systems
• Submitted for publication:
• On McMullen-like mappings (joint with Antonio Garijo) (pdf) (arxiv)
(abstract)

#### On McMullen-like mappings(joint with Antonio Garijo)

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of $f_{\lambda}$ belong to the trap door. We generalize this behavior defining a McMullen-like mapping as a rational map $f$ associated to a hyperbolic postcritically finite polynomial $P$ and a pole data $\mathcal{D}$ where we encode, basically, the location of every pole of $f$ and the local degree at each pole. In the McMullen family, the polynomial $P$ is $z\mapsto z^n$ and the pole data $\mathcal{D}$ is the pole located at the origin that maps to infinity with local degree $d$. As in the McMullen family $f_{\lambda}$, we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial $P$ and the pole data $\mathcal{D}$. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery.

• In progress:
• Trees associated with the configuration of postcritically separating Julia components
(abstract)

#### Trees associated with the configuration of postcritically separating Julia components

This paper explains how some particular classes of non post-critically finite rational maps may be characterized by the data of a weighted Hubbard tree together with holomorphic models for every branching point of the tree. A general realization theorem of such combinatorial datas in the hyperbolic case is proved. This result is illustrated by the existence proof of a family of degree 4 rational maps with a periodic cycle of buried Julia components of arbitrary large period. In particular, the first return map is of degree arbitrary large, that can not hold in the polynomial case. Finally, the non-hyperbolic case is discussed.

• A Levy's criterion for non post-critically finite maps
(abstract)

#### A Levy's criterion for non post-critically finite maps

Although difficult, Thurston's combinatorial characterization of post-critically finite rational maps is major tool in complex dynamics. Levy's works allow to simplify this criterion in the polynomial case. This paper provides a new proof of this result that extends to non post-critically finite polynomials. This generalization is illustrated by a sufficient condition for existence of polynomials with a fixed Siegel disk of bounded type.

• Wandering domains for composition of entire functions
(abstract)

#### Wandering domains for composition of entire functions

In this joint article with Núria Fagella and Xavier Jarque, we use a recent result of Bishop to construct new examples of transcendental entire functions admitting wandering Fatou domains. In particular, we show that a composition of two entire functions without wandering Fatou domains may have a wandering domain.

Some talks:

### Teaching activities:

I gather here my lecture notes, worksheets, examination problems and solutions (archives open when the mouse hovers over the years).

• 2011-2012
2011-2012
• L1 Mathematics (MI, English-speaking students), 2nd semester
• L1 Mathématiques pour les sciences (PCST), 2ème semestre
• Leçons de préparation à l'Agrégation externe de Mathématiques
• 2010-2011
2010-2011
• L1 Mathematics (MI, English-speaking students), 2nd semester
• L1 Mathématiques pour les sciences (PCST), 2ème semestre
• Leçons de préparation à l'Agrégation externe de Mathématiques
• 2009-2010
2009-2010
• L1 Mathématiques pour les sciences (SV), 1er semestre
• L2 Algèbre linéaire et bilinéaire
• L3 Compléments d'analyse
• Leçons de préparation à l'Agrégation externe de Mathématiques
• 2008-2009
2008-2009
• L1 Mathématiques pour les sciences (PCST), 1er semestre
• L1 Mathématiques pour les sciences (SV), 1er semestre
• L3 Compléments d'analyse
• 2007-2008
2007-2008
• L2 Analyse dans R^n
• Leçons de préparation à l'Agrégation externe de Mathématiques
• 2006-2007
2006-2007
• L1 Mathématiques pour les sciences (MI), 2ème semestre
• L1 Mathématiques pour les sciences (SV), 1er semestre

Some useful files:

### Popular sciences:

• Introductory talk (in French) about the logistic map and chaos theory (for SMF)
• Talk (in French) about the brachistochrone curve for high school classes (for Animath) in 2012
• Talk (in French) about the Riemann zeta function for high school classes (for Animath) in 2011
• List of problems (in French) for MATh.en.JEANS groups (workshops in Junior schools) in 2007-2008
• List of problems (in French) for MATh.en.JEANS groups (workshops in Junior schools) in 2006-2007
• Poster (in French) about fractal geometry and teaching-research jobs (for the week of research and innovation in Val d'Oise in 2007)