## IHP

Institut Henri Poincaré

## Julia sets, snowflakes and distortion of dimension under a holomorphic motion of a circle

Type:
Type:
Site:
Date:
10/10/2014 - 10:30
Salle:
421
Orateur:
Kari Astala
Localisation:
Université d'Helsinki
Localisation:
Finlande
Résumé:

Holomorphic motions provide a bridge between analysis and complex dynamics, and provide also powerful tools for the quasiconformal mappings. Basic examples of holomorphic motions are given by Julia sets when parameters of polynomials are varied, and these give important examples within different topics in analysis, as well.

In this talk, based on a joint work with Ivrii, Prause and Perälä, we are interested in maximal growth of dimension under holomorphic motions of Julia sets. In particular, consider the family $P(z) = z^d + tz$ with $|t| <1$. Slodkowski's theorem allows for the Böttcher coordinates a natural extension to a holomorphic motion of the plane. But can there exist a better one ?

## Fast basins, fractal manifolds, and flows on attractors of iterated function systems (IFSs)

Type:
Type:
Site:
Date:
06/06/2014 - 10:30
Salle:
421
Orateur:
Michael Barnsley
Localisation:
Université nationale australienne
Localisation:
Australie
Résumé:

Since their introduction, thirty years ago, IFSs have become a widely used concept. Reasons for this success include the simplicity of the IFSs themselves and the richness of their attractors. This talk will define, characterize, and exemplify, a number of new basic structures that arise naturally from an attractor of an IFS. The driving force behind these structures is a direct generalization of the notion of analytic continuation from smooth to rough.

## On the Hausdorff Dimension of the quadratic Julia sets near to the parabolic points

Type:
Type:
Site:
Date:
16/05/2014 - 10:30
Salle:
314
Orateur:
Ludwik Jaksztas
Localisation:
Université de Varsovie
Localisation:
Pologne
Résumé:

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).

## On McMullen-like mappings

Type:
Type:
Site:
Date:
04/04/2014 - 10:30
Salle:
421
Orateur:
Sébastien Godillon
Résumé:

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.

## Thermodynamical formalism and expanding Thurston maps

Type:
Type:
Site:
Date:
21/03/2014 - 10:30
Salle:
421
Orateur:
Zhiqiang Li
Localisation:
Université de Californie, Los Angeles
Localisation:
États-Unis
Résumé:

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.

## Des propriétés typiques d'une famille de transformations du cylindre $\mathbb T^1\times \mathbb Z$

Type:
Type:
Site:
Date:
07/03/2014 - 10:30
Salle:
421
Orateur:
Alba Málaga
Localisation:
Université Paris 11
Localisation:
France
Résumé:

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.

## CATTIAUX Patrick

Date:
Mer, 05/02/2014 - Jeu, 06/02/2014
Site:
Nom:
CATTIAUX
Prénom:
Patrick
Origine:
Université Toulouse 3
Origine:
France
Thème:
Exposé séminaire C'TOP
Invitant:
GOZLAN Nathaël

## Rigidity for Sierpinski carpet Julia sets

Type:
Type:
Site:
Date:
14/02/2014 - 10:00
Salle:
421
Orateur:
Mario Bonk
Localisation:
Université de Californie, Los Angeles
Localisation:
États-Unis
Résumé:

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.

## No unexpected wandering domains for Bishop's example

Type:
Type:
Site:
Date:
07/02/2014 - 10:30
Salle:
421
Orateur:
Sébastien Godillon
Résumé:

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.

## Measure and Hausdorff dimension of randomized Weierstrass-type functions

Type:
Type:
Site:
Date:
17/01/2014 - 10:30
Salle:
421
Orateur:
Julia Romanowska
Localisation:
Université de Varsovie
Localisation:
Pologne
Résumé:

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 $ab>1$.