Schedule

The lunch (at noon) of registered participants will be covered by the organisers and will take place at Mas de Dardagna.

Abstracts and slides

The space of closed sub semigroups of ℂ* together with the point at infinity has naturally the topology induced by the Hausdorff topology on the space of compact subsets of the Riemann sphere. I will give a description of the closure of the space of one generated sub semigroups. Application : given a polynomial f with a Siegel cycle, a LLC map is a map defined on an open subset of the iterated preimages of the Siegel disk cycle, which takes values in an open subset of the cycle itself and such that for each connected component of its domain, it induces, via the linearizing coordinate, a linear map (depending on the component) between open subsets of the unit disk. I will show that Douady-Epstein enrichments of the dynamic [f] with domain of definition in the iterated preimages of the cycle are precisely LLC maps.

We investigate the set of quadratic rational maps which possess a degenerate parabolic fixed point.

I will explain the first steps in an attempt to define an invariant class under near parabolic renormalization for z^d+c.

The notion of polynomial-like mappings was introduced by Douady and Hubbard in the ground-breaking paper 'On the dynamics of Polynomial.like mappings' (1985). It has been proven to be instrumental in understanding and solving a host of problems in holomorphic dynamics. A polynomial-like mapping of degree d is naturally characterized by two disjoint sub-dynamical systems called the internal class and the external class. The external class is a degree d orientation preserving, strongly expanding (hence hyperbolic) covering of the unit circle by itself. We consider a new class of maps similar to polynomial-like mappings but with the external map only weakly expanding, i.e., with parabolic periodic points, and we will call them parabolic-like maps. Since the parabolic periodic points for the external map attract points from the complement of the unit circle, the filled Julia set is not an outward repeller and the domain of such a map can not be relatively compact in the range.

A holomorphic correspondence from the Riemann sphere to itself is defined by an algebraic equation in two complex variables. When iterated, holomorphic correspondences generalise rational maps and finitely generated subgroups of PSL(2,ℂ). We consider mixed (forward and backward) iteration and the question of equi-continuity. An invariant (admissible) conformal metric is known to exist on the Fatou domains (minus the grand orbits of attracting periodic cycles) of a rational map and is conjectured to be uniformly continuous with respect to the spherical metric. Such an invariant conformal metric is uniformly continuous on "orbi-compact" subsets. A similar result holds for a domain fully invariant under a correspondence where the action lifts to a group resolution.

Thurston's theorem is about self branched coverings of 2-sphere and describes when it can be equivalent to a rational map. His condition involves infinitely many systems of simple closed curves and in general it is very difficult to check. Previous successful cases are topological polynomials and mating of quadratic polynomials. In this talk, we try to give another case which is constructed by plumbing from a piecewise linear map on a tree. We will see how one can take advantage of an invariant multicurve which is not a Thurston obstruction.

The aim of this talk is to generalize Thurston's Theorem to postcritically infinite case, and characterize rational maps with attracting cycles, Siegel disks and Herman rings. Based on Guizhen Cui and Tan Lei's work on characterization of hyperbolic rational maps and Shishikura's Herman-Siegel surgery, we show that the combinatorics and rational realization of every non-parabolic branched covering is essentially determined by finitely many Siegel maps or Thurston maps. As an application, we give a characterization of a class of rational maps with Herman ring.