Abstracts of Talks

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Polynomial dynamics on the basin of infinity

Kevin Pilgrim

The restriction of a polynomial f to its basin of infinity X(f) yields a holomorphic dynamical system. If, for example, all critical points of f lie in X(f), then f is determined up to affine conjugacy by its restriction to X(f). Motivated by this, we investigate the classification of basin dynamical systems f: X(f) → X(f).
This is joint ongoing work with Laura DeMarco.

Geometric Limits and Renormalization

Jeremy Kahn

In joint work with M. Lyubich, we consider the problem of proving a priori bounds for an infinitely renormalizable quadratic polynomial for which the renormalization types converge to a geometric limit. In particular, we show such a bound for the case where each renormalization type is, for some q, the unique parameter in the 1/q limb where the critical point is periodic of period q + 1. We then describe the difficulty that arises in the more general case, and outline an approach to its solution.

A separation Theorem for entire transcendental functions

Nuria Fagella

Let f be an entire transcendental map in class B and of finite order (or a finite composition of finite order functions). For this class of maps we prove a separation theorem analogous to the Goldberg-Milnor-Kiwi separation theorem for polynomials. Such results have several applications, as for example that there cannot be any Cremer periodic point in the boundary of a periodic Siegel disk.
This is joint work with Anna Benini.

On the connectivity of the escaping set for complex exponential Misiurewicz parameters

Xavier Jarque

Let E_λ(z)= λexp(z),λ ∈ C be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed λ, the set of points in C with orbit tending to infinity is called the escaping set. We prove that the escaping set for all λ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set.

Nonlanding hairs in transcendental entire dynamics

Monica Moreno Rocha

The Julia set of a transcendental entire map contains path connected components known as "hairs" or "dynamical rays". Each hair is a curve extending to infinity in one direction and limiting at a single point of the plane in the other direction (so the hair "lands" at its endpoint). We are interested in studying nonlading hairs, that is, curves also extending to infinity but with a nontrivial continuum as limit set in the other direction. All known results of nonlanding hairs have been derived from the exponential family z → λe^z under the assumption that its asymptotic value is either preperiodic to a repelling cycle or escapes to infinity (and thus the Julia set is the whole complex plane). It is also known that the closure of the nonlanding hair is an indecomposable continuum (Devaney & Jarque, Rempe, MR, among others). In this talk I will present the main ideas to construct nonlanding hairs for a different family of transcendental entire maps with escaping asymptotic value. In contrast with the exponential case, the Fatou set is nonempty and the closure of each nonlanding hair is an indecomposable continuum with a distinguished point playing, in a rough sense, the role of an endpoint.
This is a joint work with A. Garijo and X. Jarque.

Rational surfaces with a large automorphism group

Serge Cantat

Let X be a surface obtained by blowing up a finite number of points of the projective plane. Let G_X be the group of its holomorphic diffeomorphisms. How large can the group G_X be? We shall describe recent results and constructions related to this problem. This involves both dynamics and algebraic geometry.

Combinatorial models of expanding maps and Julia sets

Volodia Nekrashevych

We will show a new rigidity theorem for expanding dynamical systems, and use it to describe a method of constructing a recurrent procedure producing approximations of expanding dynamical systems and their Julia sets by simplicial complexes. These recurrent procedures are generalizations of Hubbard trees, subdivision rules, and automata. They can be applied to dynamical systems in any dimension, and can be used to describe Julia sets that can not be visualized by other means. Some examples from dynamics of several complex variables will be presented.

Towards Arithmetic Thurston Rigidity

Adam Epstein

Infinitesimal Thurston rigidity is a purely algebraic statement about postcritically finite rational maps. The only known general proof is transcendental. Recent work of Patrick Ingram, and of my own, gives a proof for polynomials. Any connection between these considerations (e.g. heights) and those of the standard proof (e.g. quadratic differentials) would be nice to find. A conceptual connection would be even nicer.

Kneading sequences and irreducibility of quadratic periodic curves

Ten Lei

The algebraic curve {(c,z), z is p-periodic for z^2+c} is known to be irreducible by Bousch (1992), Morton and Patel (1994) using algebraic methods, and by Lau-Schleicher (1994, 1998) using dynamical methods. We will present a simplified dynamical proof.

Expanding Thurston maps

Daniel Meyer

Thurston maps are the objects of Thurston's classification of rational maps among "topological rational maps". Traditionally they were used as a tool to construct rational maps. Here we study them in their own right. An additional expansion property is assumed. If such a map f has an invariant Jordan curve containing all postcritical points, then f can be described in a simple geometric fashion. We show necessary, sufficient, as well as uniqueness results for such invariant curves. Then we show that there is always a sufficiently high iterate f n that possesses such an invariant curve. Via decompositions induced by such a curve the sphere S^2 can be equipped with a visual metric d adapted to f. Properties of the map are reflected in properties of the metric. For example (S^2,d) is quasisymmetrically equivalent to S^2 if and only if f is equivalent to a rational map, (S^2,d) is snowflake equivalent to S^2 if and only if f is equivalent to a Lattes map. There is some overlap with work by Cannon-Floyd-Perry as well as Haissinsky-Pilgrim.
This is joint work with Mario Bonk.

Yoccoz tau-function and applications

Mitsuhiro Shishikura

In 1990, Yoccoz proved that a quadratic polynomial has locally connected provided that its Julia set is connected, its periodic points are all repelling and it is only finitely renormalizable (Yoccoz parameter). He also proved that the Mandelbrot set is locally connected at those parameters. Ingredients of the proof are Yoccoz puzzles and parapuzzles and tau-function, which is a combinatorial tool to analyze recurrence of the critical point. Although puzzles and parapuzzles are widely used in complex dynamics, the part using tau-function has been typically replaced by Branner-Hubbard Tablaux in published papers. In this talk, we will give a brief introduction to tau-function approach. We also discuss its applications. For example, it can be used to prove that the Julia sets for Yoccoz parameters and also the set of Yoccoz parameters have Area zero; weak Jacobson's theorem for real maps; the rigidity of infinitely renormalizable real quadratic polynomials (Lyubich, Graczyk-Swiatek theorem). For the rigidity, we need to use the results on complex a priori bounds and Teichmueller theory.

Workshop webpage