Schedule

Abstracts

In this talk I present old and new results of mating polynomials. In the old result (together with M. Yampolsky), we proved existence and uniqueness of matings between any non-renormalisable polynomial f_c not in the 1/2-limb of the Mandelbrot set and the basilica polynomial p(z)=z²-1. The main idea here is to construct a bubble puzzle for a family of candidate rational maps R_a (where a is a parameter). With this bubble puzzle one can transfer the combinatorics from f_c to R_a. As we shall see, this combinatorial setup for R_a is an effcient way of proving the existence of a mating.
For degree 3 polynomials I present a result together with P. Roesch where we use a different puzzle. Our result is existence of matings between the so called double-basilica f(z)=z(z²+3/2) and a polynomial in the family f_a(z) = z²(z+3a/2) (where a is a parameter). The resulting map is a Newton map. To prove existence of matings here, we use a puzzle from the thesis of P. Roesch and a special puzzle for Newton maps to transfer combinatorics from f_a to the Newton map. Again, this combinatorial information for the Newton map will be used to prove that it is indeed a mating.

I will present questions and conjectures on "matings of polynomials" collected during previous years and also during the workshop.

A holomorphic correspondence on the Riemann sphere is a multivalued map z→w defined by a polynomial relation p(z,w)=0. There is a one complex parameter family of holomorphic correspondences which includes matings between quadratic polynomials and the modular group. In the parameter space is a (conjectured) copy of the Mandelbrot set, surrounded by a region where the correspondence acts "discretely" on the Riemann sphere, with limit set a Cantor set. We investigate the structure of this region and its boundary.

I will illustrate and partially explain how the slow procedure degenerates in this particular case

Yampolsky and Zakeri have proved that for any pair α and β of bounded type rotation numbers, the quadratic rational which fix the origin with rotation number α and infinity with rotation number β are the mating of the two quadratic polynomials with a fixed point of rotation number respectively α and β. Their proof is based on a quasiconformal surjery, which uses a Blaschke fraction as a premodel. To extend this result to almost all rotation numbers with the method of Petersen and Zakery, we need a new class of premodels, that preserve two circles. I will explain several constructions of such a class.

Abstract: A holomorphic correspondence is a multivalued map from Ĉ to itself defined by p(z,w)=0, where p(z,w) is a polynomial in two variables. Holomorphic correspondencves are of interest as they provide a setting in which to regard other dynamical systems. In particular the dynamics of both polynomials in one variable and of finitely generated Kleinian groups can be realised as correspondences. We present examples of correspondences which exhibit a new combination of polynomial-like and group-like behaviour.

According to Milnor, the mating operation is interesting because it has none of the usual good properties. It is not injective, surjective, everywhere defined, or continuous. We give a survey of discontinuity mechanisms, with special attention to the quadratic case.

In Sullivan's dictionary comparing Kleinian groups and holomorphic dynamics, what corresponds to the Tan Lei-Rees theorem on matings is the double limit theorem.
I will explain the similarities between the statements, and bring up some similarities in the proofs.

We use moduli space maps (arising from Thurston's theorem), the geometry of their periodic cycles, and Bottcher coordinates to study matings and tunings.

The Thurston characterization and rigidity theorem gives a beautiful description of when a postcritically finite topological branched cover mapping the 2-sphere to itself is Thurston equivalent to a rational function. However, Thurston equivalence remains mysterious, and it took two decades to solve the "Twisted Rabbit Problem" which sought to identify the Thurston class of the rabbit polynomial composed with an arbitrary Dehn twist. I will present an invariant for Thurston equivalence that has successfully solved the twisting problem for a degree 3 rational function.

I will present a sufficient conditions for a rational map to arise as a mating. One obtains an explicit combinatorial algorithm to determine the polynomials which yield the map when mated. More precisely one obtains the critical portraits (in the sense of Poirier) of the polynomials. This can be used to illustrate different types of shared matings. These methods are set up to deal with rational maps whose Julia-set is the whole sphere.

In this talk I will set the scene for the workshop. This includes Julia and filled Julia set. External rays and the Caratheodory loop. Then I will discuss mating definitions, discuss their properties and show some exaples. I will end by introducing Thurstons theorem including the definitions needed to understand its statement.

Thurston's characterization theorem identifies the obstructions to a critically finite branched map f: S² → S² to be equivalent to a rational function. A priori, these obstructions could lie anywhere. I will discuss two instances where intersection theory can be used to restrict the range of possibilities for obstructions. The first generalizes ideas of Shishikura and Tan and relies on arguments from linear algebra. The second relies on the idea that on any Riemann surface, the reciprocal of the intersection number of two curve families gives an upper bound on the product of their moduli.

I shall discuss a set of Wittner captures by the aeroplane. This set, which depends on n, has 2ⁿ elements, and the members of the set have critical point of preperiod O(2ⁿ). This set extends the set of n+1 Wittner captures (ETDS 2010) which are all Thurston equivalent to the same rational map. The set arises in work to extend the property of multiple equivalent Wittner captures to sets of positive density.

Suppose F is a bicritical rational map for which the two critical points belong to the attracting basins of disjoint periodic cycles. Then a cluster point for F is a point in J(F) which is the endpoint of the angle 0 internal rays of at least one critical orbit Fatou component from each of the two critical cycles. We will show that there is a set of combinatorial data that can be used to describe rational maps with cluster cycles, and that in very simple cases this combinatorial data is enough to classify the rational maps in the sense of Thurston. We also show that all rational maps with clusters can be obtained by the mating of two (unicritical) polynomials, and discuss what properties can be found for the two maps in the matings. Later on, we will discuss the difficulties in extending the results obtained so far to more complicated cases, and show how the study of matings has lead to some perhaps surprising ways of constructing rational maps with cluster cycles.

For polynomials, external angles describe the dynamics on the Julia sets. For general rational maps or Thurston maps that are not polynomials, there are no unified way of describing the dynamics. Under the presence of multiply connected Fatou components (except parabolic basins), one can define a tree and a piecewise linear map on it so that it describes the configuration of the components and use surgery to extract further information. It is also possible to construct a rational maps from such a tree map together with local models (rational maps on multiple spheres). Similarly, for Thurston maps with an invariant multicurve, one can associate a tree with a piecewise linear map. A converse construction is also possible and this yielded a first non-Levy cycle obstruction with Tan Lei. We will define unweighted Thurston matrix and effective Thurston matrix for and an estimate on their eigenvalues lead to a construction of rational maps.

This one-parameter family of cubic rational maps is one of the rare families that has a complete description in terms of matings (for the postcritically finite maps). We will illustrate in this setting how to recognize whether a mating is a rational map and vice visa, and how to build the semi-conjugacy following Rees-Shishikura.

We will discuss matings that appear on the boundaries of type C hyperbolic components in spaces of quadratic rational functions.