# Frontiers in Complex Dynamics (Celebrating John Milnor's 80th birthday) (11w5076)

## Objectives

The main objectives of the conference are:

- To foster interaction among researchers in holomorphic dynamics and allied fields such as several complex variables, Teichmuller theory, self-similar groups, arithmetic dynamics, symbolic dynamics, hyperbolic and algebraic geometry, statistical physics, etc.

- To use this interaction for designing new mathematical tools that can aid in the progress in holomorphic dynamics, especially in understanding and building models of multi-dimensional parameter spaces.

- To involve graduate students and younger researchers into discussions with more senior mathematicians. This interaction will be beneficial for their scientific growth, and will inject new fresh ideas into the field.

- To celebrate John Milnor's 80th birthday: his influence on the field of holomorphic dynamics cannot be overestimated.

Because of the great interest in and importance of this field, which continues to pose many challenging open problems, we anticipate having about 130 participants at the Workshop. Consequently, we ask the Workshop to be jointly hosted by BIRS and the Banff Conference Center, accommodating 42 participants in BIRS with the remainder hosted by the Banff Conference Center. We hope to obtain separate support for participants staying at the Banff Center. The Banff Center will provide an auditorium with sufficient capacity for the main lectures, as well as food and lodging for participants not housed at BIRS.

We anticipate that the participants staying at BIRS will be primarily senior

researchers, with the majority of the junior participants staying in the Banff Center.

The following senior researchers from the enclosed list have indicated very strong support of the idea of the conference and strong interest to participate in it, if approved. In alphabetical order by last name, they are:

Eric Bedford (Indiana University),

Bodil Branner(Technical University of Denmark),

Arnaud Cheritat(Universite Paul Sabatier Toulouse),

Laura DeMarco(University of Illinois at Chicago),

Bob Devaney(Boston University),

Bill Goldman(University of Maryland),

Vincent Guedj(Universite Aix-Marseille),

John Hubbard(Cornell University & Universite Aix-Marseille),

Jeremy Kahn(Stony Brook University),

Curt McMullen(Harvard University),

Volodymyr Nekrashevych(Texas A&M University),

Mitsuhiro Shishikura(Kyoto University),

John Smillie(Cornell University),

Dennis Sullivan (CUNY & Stony Brook University),

Tan Lei(Universite d'Angers),

Alberto Verjovsky(Instituto de Matematicas, Cuernavaca),

Misha Yampolsky(University of Toronto),

Jean-Christophe Yoccoz(College de France).

We now give a brief overview of the mathematical objectives targeted by this proposal.

Several complex variables: dynamics and parameter spaces.

Holomorphic dynamics in one variable has seen great progress in the last century. Higher-dimensional dynamics is not so advanced; extending results from one to many variables requires new techniques and approaches, and in many cases, the emerging picture is substantially different. Impressive groundwork in the higher-dimensional complex dynamics has been laid over the past 20 years by Milnor, Hubbard, Bedford and Smillie, Fornaess and Sibony, followed by many others; much remains to be done.

For example, the Henon family is the natural two-dimensional generalization of the one-dimensional quadratic family. In this context, one can define the concepts of the Fatou and Julia sets. Basic properties of these sets can be understood using techniques of pluripotential theory and non-uniformly hyperbolic dynamics. However, finer classification of these maps, combinatorial and geometric, is very difficult.

Important steps in this direction have been recently made in Cornell, towards understanding of the shift locus in the Henon family, and in Stony Brook, towards understanding renormalization. There is every reason to expect a rich theory just waiting to be uncovered, with some similarities but also significant differences from the one-dimensional model.

In addition, techniques from several complex variables, algebraic geometry, and arithmetic dynamics have proven very useful in tackling problems in one variable, for instance, in understanding two-dimensional parameter spaces of cubic polynomials and quadratic rational maps. Strengthening interactions between these fields is highly promising.

Renormalization, Local Connectivity and Area of Julia sets.

Two central themes in holomorphic dynamics, the MLC conjecture (on the local connectivity of the Mandelbrot set) and the problem of measure and Hausdorff dimension of Julia sets, are intimately related to the Renormalization Theory.

For instance, in the early 1990s Yoccoz proved local connectivity of the Mandelbrot set at all points that are not infinitely renormalizable. Recent work of Kahn and Lyubich introduced new analytical techniques that allowed them to extend Yoccoz's results to unicritical maps of arbitrary degree and to prove MLC for many infinitely renormalizable parameters. Further progress in this direction is under way. It may lead to full combinatorial classification of polynomial dynamics.

However, in rational dynamics, an adequate combinatorial framework is not yet available. For instance, the Yoccoz puzzle techniques has been developed only for some special families of rational functions. Interesting progress has been made in terms of captures and matings (stemming from the work of Milnor and Rees). The techniques of self-similar groups may be very relevant in forming a comprehensive picture of the parameter spaces. Recent work of Inou, Roesch, Kiwi, Shishikura, and others generalizing Thurston's theorem to greater classes of maps are very promising steps in this direction.

Carrying out Douady's program, Buff and Cheritat have recently constructed the first example of a Julia set of positive area, resolving a long-standing open problem. The solution makes use of three renormalization theories, at the same time enhancing them with new ideas and techniques. As usual, solution of a big problem opens new routes of research.

Here is a sample of very interesting open questions.

How big is the set of parameters with Julia sets of positive measure --

do they have Hausdorff dimension less than two? (Recall that Shishikura proved in 1990's that the boundary of the Mandelbrot set has Hausdorff dimension two).

All examples of Julia sets of positive area so far are not locally connected. Are there locally connected examples? What is the measurable dynamics on such Julia sets - is it ergodic?

We anticipate that these problems and others will certainly be a

good part of discussions at the Workshop.

Iterated Monodromy Groups and Laminations.

Iterated monodromy groups, recently introduced to holomorphic dynamics by Nekrashevych, opened a fresh direction of research in the field. It is closely related to the laminations earlier constructed by Lyubich and Minsky. Deeper understanding of these objects in their interplay, from algebraic and geometric points of view, is a very interesting ongoing activity.

Already the simplest maps like $z^2 -1$ and $z^2 + i$ show that the Iterated Monodromy Groups and the corresponding laminations can have a very intricate structure, but are also amenable to study by combined methods of geometric group theory and holomorphic dynamics.

Iterated Monodromy

Groups have already found applications to the dynamics of post-critically finite polynomials and rational functions, with important links to combinatorial equivalences of self-coverings of the Riemann sphere and Thurston's theorem. An example of the power of this machinery is that it has been used to solve difficult questions in holomorphic dynamics such as Hubbard's "twisted rabbit" problem. It has also been used by Sarah Koch to produce many examples of post-critically-finite maps in several complex variables.

Laminations bring a tight explicit link between holomorphic dynamics and

hyperbolic geometry, providing new insights into the rigidity problem. They also serve as a link between one- and two-dimensional objects, as Julia sets of H'enon maps degenerate to laminations as the Jacobian of the map vanishes. No doubt, self-similar groups and laminations will provide further illuminating insights into the intricate structure of objects produced by dynamics.

We are certain that a conference centered around the above themes would be an exciting scientific event that would attract many leading experts in different fields along with many young researchers, would stimulate new ideas and open new avenues of research. We look forward to an opportunity to organize such a Workshop in the beautiful Banff setting.