Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial z^2+c. I will discuss the behaviour (derivative) of the function d(c) where the real parameter c is close to the parabolic points, especially 1/4 (parabolic point with one petal) and -3/4 (point with two petals).

McMullen has proved that the Julia set of $z^n+\lambda/z^d$ is a Cantor of Jordan curves as soon as the local degrees $n$ and $d$ satisfy a certain arithmetic condition (and $|\lambda|>0$ is small enough). Many other authors have studied similar examples obtained by adding singular perturbations to a polynomial. I will introduce a general definition of the so called McMullen-like mappings that unifies this behavior. Every topological conjugation class is described by the data of a postcritically finite hyperbolic polynomial and a collection of local degrees, each associated with a periodic Fatou domain. Using this invariant, the arithmetic condition for existence can be generalized. I will show that this condition is actually necessary by using the theory of Thurston's obstructions.

Thurston maps are a class of branched covering maps on the 2-sphere that arose in W. Thurston's characterization of postcritically finite rational maps. By imposing a natural expansion condition, M. Bonk and D. Meyer investigated a subclass of Thurston maps known as expanding Thurston maps, which turned out to enjoy nice topological, metric, and dynamical properties.

Thermodynamical formalism has been a powerful tool, for many classical dynamical systems, to investigate invariant measures whose Jacobian functions have strong regularity properties.

In this talk, we will first introduce expanding Thurston maps with some motivation from their connection to other topics of mathematics. We will then use thermodynamical formalism to sketch a proof for the existence, uniqueness, and exactness of equilibrium states for expanding Thurston maps and Hölder continuous potentials.

If time permits, we will also show that an expanding Thurston map is asymptotically $h$-expansive if and only if it has no periodic critical points, which suggests the subtlety of our notion of expansion.

Je vais présenter mon travail sur une famille de systèmes dynamiques qui est heuristiquement liée à un billard sur un parallélogramme. Cette famille est définie sur le cylindre discret $\mathbb T^1\times \mathbb Z$ où $\mathbb T^1={\mathbb R/\mathbb Z}$ est le tore unidimensionnel (c'est à dire le cercle). Pour toute suite bi-infinie $\underline\alpha\in\mathbb T^\mathbb Z$, nous définissons la transformation $F_{\underline\alpha}$ presque partout sur ​​le cylindre comme suit:
$$F_{\underline\alpha}\left([x]_\mathbb
Z,n\right)=\left([x+\alpha_n]_\mathbb Z, n+\left\{\begin{array}{rcl} 1 & if
& x+\alpha_n\in (0,\frac12)+\mathbb Z\\ -1 & if & x+\alpha_n\in
(-\frac12,0)+\mathbb Z\\ \end{array}\right. \right).$$
Quand la suite $\underline\alpha$ est constante et irrationnelle, Conze et Keane ont montré que $F_{\underline\alpha}$ est ergodique. J'essaie de comprendre ce que sont les propriétés typiques de $F_{\underline\alpha}$ dans un certain sens. À savoir, quelles sont les propriétés satisfaites pour presque tout $\underline\alpha$ ou pour $\underline\alpha$ générique dans l'espace des paramètres? Pour le moment, j'ai prouvé que la conservativité est à la fois générique et presque sure, alors que la minimalité est générique. Je voudrais comprendre également les propriétés de diffusion de cette famille.

Sierpinski carpets are interesting fractals that can arise in a dynamical setting as limits sets of Kleinian groups or Julia sets of rational maps. While the topology of Sierpinski carpets is well-understood, difficult problems arise if one investigate their quasiconformal geometry. In my talk I will focus on some recent joint work with M. Lyubich and S. Merenkov on quasiconformal rigidity questions for Sierpinski carpets that are Julia sets of postcritically-finite rational maps.

The construction of transcendental entire functions by quasi-conformal foldings recently provided by Bishop has allowed him to produce the first known example of wandering domain in Eremenko-Lyubich's class. After explaining Bishop's strategy, the main purpose of the talk is to show how a recent result of Lasse Rempe and Helena Mihaljevic-Brandt of hyperbolic geometry in transcendental dynamics may be applied to prove that Bishop's example has no other wandering domain than those expected.

In my talk I will consider functions of the type $$f(x) = \sum_{n=0}^\infty a_n g(b_nx+\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \geq b >1$, $a^2b> 1$ and $g$ is a $C^1$ periodic real function with finite number of critical points in every bounded interval. I will prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D = 2+ \log{a}/\log{b}$ provided $ b = \lim_{n\rightarrow \infty}b_{n+1}/b_n>1$ and $ab>1$.

The problem of characterizing the algebraic numbers arising from dynamical systems has recently drawn considerable attention.

One of the first contexts in which this question makes sense is in the family of real quadratic polynomials; in this case, W. Thurston considered the set of all Galois conjugates of entropies of real quadratic polynomials with finite critical orbit, and found out that this set displays an extremely rich fractal structure.

We shall prove that such a set (or rather, its closure) is path-connected and locally connected. Pictures will be provided.

Les nombres d'Hurwitz comptent des classes d'équivalence de certains revêtements ramifiés de la sphère par elle-même, ou autrement dit, de fonctions rationnelles. Comme Hurwitz l'avait déjà vu à la fin du 19e siecle, on peut représenter ces objects combinatoirement par des factorisations dans le groupe symétrique, ou par des plongements de graphes dans des surfaces, aussi appelés "cartes". Je montrerais comment ce dernier point de vue permet, en s'appuyant sur une théorie combinatoire du codage des cartes par des arbres, de retrouver bijectivement les formules d'Hurwitz et d'en obtenir de nouvelles extensions pour les nombres de Hurwitz doubles.