A well studied type of non-uniformly expanding maps are the Collet-Eckmann maps (CE), introduced in the 1980s by P. Collet and J-P. Eckmann. These maps has positive Lebesgue measure in the space R_d of rational functions for any degree d ³ 2, which was shown in my thesis in 2004. A special type of CE-maps are the so-called Misiurewicz maps, which have no parabolic cycles and the critical set on the Julia set is non-recurrent (they are also assumed to be non-hyperbolic). These maps have Lebesgue measure zero but full Hausdorff dimension in R_d for any d ³ 2. Moreover, every Misiurewicz map can be approximated by a hyperbolic map (answering a conjecture by Herman). Some results of this type also extends to semi-hyperbolic maps, which do not have any parabolic cycles and for which every critical point on the Julia set is non-recurrent (instead of the whole critical set).

Within the family P_d of degree d polynomials, we investigate the distribution of the hypersurfaces Per_n(w) which consist of polynomials possessing a n-cycle of multiplier w. Using properties of Lyapunov exponents, we show that the sequence of weighted current of integration d^-n [Per_n(e^iq)] is converging to the bifurcation current T_bif for any (e^iq) ¹ 1.

We investigate the rate of convergence of the Walk on Spheres algorithm, an efficient method for computing harmonic measure.

Given l in the unit disk, we consider the family of cubic polynomials P_l,a(z) = l(z+az²+z³). There is a unique power series f_l,a(z) = z+o(z) such that f_l,a (lz) = P_l,a ( f_l,a(z) ). We let r_l(a) be the radius of convergence of the series f_l,a. The function u_l = -logr_l is continuous and subharmonic. We study the behavior of the functions u_l and the measures m_l : = Du_l as |l| ® 1.

Joint work with Arnaud Chéritat and Carsten Petersen.

I will survey recent results concerning the dynamics of the mapping class group of the four punctured sphere S²₄ on the character variety, i.e., on the space of conjugacy classes of representations of the fundamental group of S²₄ into SL₂(C). Using tools from holomorphic dynamics one can study orbits of this action and get a new viewpoint on the set of quasi-fuchsian representations.

In dynamical space C, collapsing each connected component of a level set of the escape-rate function to a point yields a tree. I will describe a parameter-space analog: if you collapse each connected component of a fiber of the critical escape-rate map, you obtain a space of "augmented trees".

We say that an infinitely renormalizable quadratic polynomial z ® z² + c has bounded-primitive type if the ratio between consecutive primitive renormalization times is bounded. We prove the a priori bounds for bounded-primitive type maps, which establishes the local connectivity of the Mandelbrot set at the corresponding parameter values.

We present joint work with M. Urbanski on ergodic properties of sub-hyperbolic meromorphic functions with polynomial Schwarzian derivative and, in particular, establish the existence, ergodicity and unicity of a conformal measure with minimal exponent h together with a Bowen's formula showing that this number h equals the Hausdorff dimension of the Julia set of the function. So, as expected, the theory of these sub-hyperbolic functions seems to behave like the theory of hyperbolic functions. However, it turns out that there is a significative difference: there always exists an invariant measure which is absolutely continuous with respect to the above conformal measure but we will see that, in many cases, this measure is only s-finite.

We discuss properties of the so-called Thurston pullback map on Teichmueller space associated to a postcritically finite branched covering of the two-sphere to itself.

This is joint work with X. Buff, A. Epstein, and S. Koch.

In a classical work, Yang and Lee proved that zeros of certain polynomials (partition functions of Ising models) always lie on the unit circle. Distribution of these zeros control phase transitions in the model. We study this distribution for a special "Migdal-Kadanoff hierarchical lattice". In this case, it can be described in terms of the dynamics of an explicit rational function in two variables. We show that the Yang-Lee zeros are organized in a transverse measure for a dynamical foliation on an invariant cylinder. From the complex point of view, they get interpreted as slices of a dynamical (1,1)-current on the projective space.

This is a joint work with Pavel Bleher and Mikhail Lyubich.

I will give an idea of the proof of the following result obtained with Yin Yongcheng.

Theorem    For a polynomial, any bounded Fatou component, which is not a Sigel disk, has a boundary which is a Jordan curve.

Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston-Teichmüller theory.