Title : Polynomial basins of infinity and the moduli space.
Speakers : Laura DeMarco and Kevin Pilgrim
1. Polynomial basins of infinity.
2. Translation surfaces from polynomials.
3. Critical escape rates and moduli spaces
4. Classification of polynomial basins
Abstract : In this series of lectures, we will present a collection of tools for studying polynomial dynamics. Our main goal is to create a map of the moduli space, to understand the components of structural stability outside the connectedness locus. The key tool is the escape-rate function on the basin of infinity. Building on the work of Branner and Hubbard on cubic polynomials (1988, 1992), we obtain a combinatorial classification of polynomial basins of infinity (up to topological conjugacy). Along the way, we explore trees and translation surfaces and quasiconformal deformations and rigidity.

Title : Rays landing at Misiurewicz and parabolic parameters.
Speaker : Tan Lei
Abstract : We use a surgery, a distortion control on univalent functions and Thurston's characterization theorem to prove landings of certain parameter rays. This is a joint work with Cui Guizhen.

Title : Limits of Quadratic-Like Maps.
Speakers : Adam L. Epstein and Carsten Lunde petersen
1. Rescalings of Blaschke products. (Epstein)
2. Secondary rescaling limits. (Petersen)
Abstract : Let \(f_k\) be a sequence of quadratic rational maps diverging in moduli space. Under certain circumstances, there may exist a sequence of conjugate maps \(F_k\) and some \(q\geq 2\) such that the iterates \(F_k^q\) converge algebraically to a rational map of degree 2 or more. It has been conjectured that, up to a suitable notion of equivalence, there can be at most two such {\em rescaling limits}, the first a quadratic rational map with a fixed point of multiplier 1, the second a quadratic polynomial.In the first talk We focus on the first rescaling limit, with attention to the special case of Blaschke products. In the second talk focus will be on the second rescaling limit.

Title : Parabolic-like mappings and correspondences .
Speaker : Luna Lomonaco
Abstract : A polynomial-like mapping is a proper holomorphic map \(f : U' \rightarrow U\), where \(U',\,\, U \approx \Bbb{D}\), and \(U'\subset \subset U\). This definition captures the behaviour of a polynomial in a neighbourhood of its filled Julia set. A polynomial-like map of degree \(d\) is determined up to holomorphic conjugacy by its internal and external classes, that is, the (conjugacy classes of) the restrictions to the filled Julia set and its complement. In particular the external class is a degree \(d\) real-analytic orientation preserving and strictly expanding self-covering of the unit circle: the expansivity of such a circle map implies that all the periodic points are repelling, and in particular not parabolic.

We extended the polynomial-like theory to a class of parabolic mappings which we called parabolic-like mappings. In this talk we present the parabolic-like mapping theory, and its uses in the family of degree \(2\) holomorphic correspondences in which matings between the quadratic family and the modular lie group.

Title : Boundary of hyperbolic components in slices
Speaker : Pascale Roesch
Abstract : Informations on the bifurcation locus at the boundary of hyperbolic components can be deduced from the informations coming from inside of those hyperbolic components when one restricts to slices. The main tool used is the persistance of internal rays and the "partition" they provide.

We will discuss 3 examples of slices:

1. the bifurcation locus is at the boundary of one hyperbolic component,
2. hyperbolic components touch through a Cantor set of boundary points,
3. boundary of hyperbolic components do not touch.

Title : Immersion of Teichmüller space into moduli space
Speaker : Matthieu Astorg
Abstract : Teichmüller theory's goal is to study deformations of the complex structure of a Riemann surface. In the 80's, McMullen and Sullivan introduced an analogue of this theory in the context of iterations of a rational map \(f\). In particular, they constructed a "dynamical Teichmüller space" which is a simply connected complex manifold, with a holomorphic map \(F\) defined on Teich(\(f\)) and taking values in the space of rational maps of the same degree as f, and whose image is exactly the quasiconformal conjugacy class of \(f\). A natural question, raised in their article, is to know if this map \(F\) is an immersion: it turns out the answer is affirmative. A. Epstein has an unpublished proof of this; we will expose a different approach.

Title : On the topological differences between the Mandelbrot set and the tricorn
Speaker : Sabyasachi Mukherjee
Abstract : We study the iteration of quadratic anti-polynomials \(\bar{z}^2 + c\) and their connectedness locus, known as the tricorn. The topology of the tricorn differs vastly from that of the Mandelbrot set (the connectedness locus of quadratic polynomials); for example, the tricorn contains parabolic arcs, which are real- analytic arcs consisting of quasi-conformally equivalent parabolic parameters and there are bifurcations between hyperbolic components along parts of such arcs. Quite recently, Hubbard and Schleicher proved that the tricorn is not path-connected, confirming a conjecture of Milnor. The goal of this talk is to prove the following results, which further elucidate the topological differences between these two parameter spaces.

1. Rational parameter rays at odd-periodic angles of the tricorn do not land at a single parameter, but accumulate on an arc of positive length in the parameter space. (This is a joint work with Hiroyuki Inou)
2. Centers of hyperbolic components are not dense on the boundary of the tricorn.

Title : Quasi-elementary correspondences
Speaker : Christopher Penrose
Abstract : A correspondence is an algebraic multi-valued function of the Riemann sphere. When iterated one can talk about the equi- continuity set. If we write \(\omega(z)\) for the accumulation set of the grand orbit of a point \(z\), we call \(w\) a ``limit'' point if \(\omega^{-1}(w)\ne\emptyset\), and a ``generic limit'' point if \(\omega^{-1}(w)\) is somewhere dense. A correspondence is elementary if its equi-continuity set excludes at most finitely many points.

A subgroup of \({\rm PSL}(2,{\bf C})\) is ``quasi-elementary'' if the complement of its equi-continuity set is contained in a round circle. Any subgroup of \({\rm PSL}(2,{\bf C})\) containing limit points in the equi-continuity set is quasi-elementary from where one can deduce that the limit set ``spreads'' to contain the whole equi-continuity set (possibly minus a couple of points). If we restrict to the ``generic limit'' set, one can propose a similar ``spreading principle'' for correspondences. To this aim we attempt to define ``quasi-elementary'' for correspondences, both generalising the notion for subgroups of \({\rm PSL}(2,{\bf C})\) and containing the class of elementary correspondences.