Minicourses

Germs of generic analytic families of diffeomorphisms unfolding a parabolic point.

By Christiane Rousseau, Université de Montréal.

Abstract: The general theme of the mini-course concerns the classification of germs of generic analytic families of diffeomorphisms unfolding a parabolic point under analytic conjugacy. The first lecture will focus of the “preparation” of a generic k-parameter family of diffeomorphisms unfolding a parabolic point of codimension k. The preparation brings the family to a prenormal form. In particular, it comprises a change of parameters to “canonical” parameters which are (almost) analytic invariants. Hence, conjugacy of families must (almost) preserve the canonical parameters and we can work for fixed values of the parameters. The last two lectures will focus on the codimension 1 case. In the second lecture we will explain the strategy of classification of families. For a fixed value of the parameter, we conjugate the diffeomorphism to the formal normal form, over two sectors whose is a neighborhood of the parabolic point. The mismatch between the two normalizations on the intersection of the two sectors is the modulus of classification: two germs of families are analytically conjugate if and only if they have the same modulus. This modulus is an unfolding of the Ecalle modulus of a parabolic point. We will describe the parametric resurgence phenomenon. The third lecture will present the space of realizable moduli. This implies understanding the dependence of the modulus on the parameter.


Geometric limits: examples and uses in Kleinian Groups and Dynamics.

By John H. Hubbard, Université de Provence and Cornell University.

Abstract: In the first talk, I will recall the classification of conjugacy classes of holomorphic germs tangent to the identity, in particular the construction of Ecalle-Voronin invariants. In the second talk, I will recall the notions of Lavaurs maps and parabolic enrichments in holomorphic dynamics. In the third talk, I will present the proof by Shishikura that the boundary of the Mandelbrot set has Hausdorff dimension 2.


Geometric limits in conformal dynamics.

By Adam L. Epstein, Warwick University.

Abstract: In the first talk, I will present the notion of geometric limits of holomorphic dynamical systems. In particular, I will present the topology associated to geomteric convergence. In the second talk, I will present the notion of analytic finite type maps, which naturally appears when studying the geometric limits of rational maps. I will then state the structure theorems asoociated to those finite type maps. In the third talk, I will present towers of analytic finite type maps.

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