THEMATIC PROGRAMS

Therefore Royden-Gardiner theorem is an obvious candidate of tools to obtain the contraction. This idea was first used by Sullivan in his work on generalized Feigenbaum type renormalization. In this talk, we will discuss two applications of Teichmuller theory to renormalizations:

1. Parabolic renormalization for parabolic fixed points and their perturbations. 2. Rigidity of real quadratic polynomials only using Yoccoz's combinatorial a priori bounds.

For these cases, we deal with the Teichmuller spaces of a disk or a punctured disk and show that a certain inclusion map induces a contraction in Teichmuller distance.

Jeremy Kahn (SUNY Stony Brook / Fields Institute)
The complex a priori bounds, continued
We describe how to reduce the a priori bounds for bounded-primitive type renormalization to a simple combinatorial statement regarding the combinatorial dynamics of arcs. We show how the combinatorial statement follows from the expanding dynamics on the Hubbard tree.

One can say (truthfully) that there is no canonical choice of fundamental domain for V_{3,m}, which is obtained from V_{3} by removing a natural dynamically defined puncture set. I shall exhibit a fundamental domain, using the dynamical planes of three maps within V_{3} and a theory known as the ``Resident's View''. This enables one to at least formulate an analogue of MLC for this family.

Three parts of the fundamental domain are straightforward (although the proof in one of these cases is probably new). The structure of the fourth part is much more interesting, involving a spiral in the dynamical plane of the so-called ``aeroplane'' quadratic polynomial z\mapsto z^{2}+c for the (unique) real parameter c for which the critical point 0 has period 3.

Roland Roeder (Fields)
Newton's Method in two complex variables: Indeterminant points and the topology of basins of attraction
The equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, but the Newton Map
N: C^2 \rightarrow C^2 for finding the four roots has very complicated dynamics: N is four-to-one and N has points of indeterminacy. Furthermore, high iterates of N have many points of indeterminacy. By restricting to parameters |1-B|>1, all of these points of indeterminacy are in the set X_l = Re(x) < 1/2, which is invariant under N. If one wants to consider the homotopy type of a basin of attraction, W(r_i), for one of the roots r_i \in X_l under N, one encounters a kind of ``topological indeterminacy.'' When studying the homotopy type of a loop \gamma in W(r_i), should one consider the homotopies that hit the points of indeterminacy of N^k or should one avoid them? Both seem reasonable. To avoid such questions, one can perform blow-ups at the points of indeterminacy of all iterates of N, obtaining a new space X_l^\infty from X_l on which all iterates of N are defined.

We show how to make precise the notion of linking numbers in X_l^\infty, overcoming a different kind of indeterminacy that is a result of the fact that H_2(X_l^\infty) is infinitely generated. Having developed this technology, I will explain how we can use it to study the homotopy type of the basins of attraction W(r_i) within X_l^\infty.

Nov. 24, 1:10 p.m.
**Note: special day and time

Robert MacKay (University of Warwick, UK)
Some robustly mixing fluid flows
Abstract: I suggest an example of a C^3 divergence-free vector field in a domain of R^3 with smooth boundary, vanishing on the boundary, which is mixing and remains so for all small perturbations in this class. Two other candidates are also presented.

Roland Roeder (Fields Institute)
Super-stable manifolds for Newton's methods in two complex variables
Abstract: While the equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, the dynamics of the Newton map N(x,y) for finding the four roots is quite complicated. In particular, N is many-to-one and N has points of indeterminacy.
The two vertical lines x=0 and x=1 are invariant under N and super-attracting. Within these lines the ''circles'' Re(y) = 1/2 and Re(y) = (1-B)/2, respectively, are hyperbolically repelling with multiplier 2. In this talk we will prove that these circles have superstable manifolds of real dimension 3 using the technique of holomorphic motions. These manifolds extend to all points with Re(x) < 1/2 and Re(x) > 1/2 respectively and provide insight into the topology of the basins of attraction for the four roots.
This work follows the ideas of John H. Hubbard and Sebastien Krief.

M. Shishikura, Kyoto University
Renormalization for irrationally indifferent fixed points of holomorphic maps
Abstract: Indifferent fixed points of holomorphic maps give rise to delicate problems such as linearization problems, discontinuous Julia sets etc. In this talk, we review the study of those phenomena from the renormalization point of view. We define a certain class of holomorphic maps with "non-degenerate" parabolic fixed points. The parabolic renormalizaion is defined for this class and shown to leave the class invariant. Then the class is still invariant under a small perturbation, therefore we can handle irrationally indifferent fixed points with large continued fraction coefficients. This is a joint work with Hiroyuki Inou.
We will compare this approach with Yoccoz's and McMullen's renormalizations. We will also mention possible applications such as Buff-Cheritat's work toward positive measure Julia sets.