The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps: the real segment is replaced by an invariant tree, known as Hubbard tree, which lives inside the filled Julia set. We prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.

En dynamique réelle uni-dimensionnelle, il existe un fort lien entre les exposants de Lyapunov des points typiques (par rapport à la mesure de Lebesgue) et l'existence de mesure de probabilité invariante et absolument continue par rapport à la mesure de Lebesgue. Ce lien persiste, on espère, en dynamique rationnelle. Pour les applications de type Misiurewicz de la famille exponentielle $z \mapsto \lambda e^z$, les exposants de Lyapunov pour des points typiques n'existent même pas.

On va décrire, pour un revêtement ramifié (par la sphère au-dessus de la sphère) postcritiquement fini $f$ à orbifold hyperbolique, un graphe fini qui, s'il existe, caractérise $f$. Cela signifie que si un deuxième revêtement ramifié $g$ possède le "même" graphe alors $f$ et $g$ sont Thurston équivalents. Si $f$ et $g$ sont des fractions rationnelles cela conduit à une conjugaison par une transformation de Möbius. On va construire de tels graphes pour certains accouplements et on verra un exemple d'utilisation.

We will see in examples how univalent maps arise as pseudo-conjugacies in a dynamical perturbation of a rational map, how to control their conformal and spherical distortions and how to use these controls to get ray-landing properties in the parameter space of rational maps.

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 ?

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.