Quadratic polynomials have been investigated since the beginnings of complex dynamics, and are often approached through combinatorial theories such as laminations or Hubbard trees. I will explain how both of these approaches fit in a more algebraic framework: that of iterated monodromy groups. The invariant associated with a quadratic polynomial is a group acting on the infinite binary tree, these groups are interesting in their own right, and provide insight and structure to complex dynamics: I will explain in particular how the conversion between Hubbard trees and external angles amounts to a change of basis, how the limbs and wakes may be defined in the language of group theory, and present a model of the Mandelbrot set consisting of groups. This is joint work with Dzmitry Dudko and Volodymyr Nekrashevych.[-]

I will introduce the Speiser and Eremenko-Lyubich classes of transcendental entire functions and give a brief review of quasiconformal maps and the measurable Riemann mapping theorem. I will then discuss tracts and models for the Eremenko-Lyubich class and state the theorem that all topological tracts can occur in this class. A more limited result for the Speiser class will also be given. I will then discuss some applications of these ideas, focusing on recent work with Kirill Lazebnik (prescribing postsingular orbits of meromorphic functions) and Lasse Rempe (equilateral triangulations of Riemann surfaces).[-]

Fatou noticed in 1926 that certain transcendental entire functions have Julia sets in which there are curves of points that escape to infinity under iteration and he wondered whether this might hold for a more general class of functions. In 1989, Eremenko carried out an investigation of the escaping set of a transcendental entire function f, $I(f)=\left \{ z\in\mathbb{C}:\left | f^{n}\left ( z \right ) \right | \rightarrow \infty \right \}$ and produced a conjecture with a weak and a strong form. The strong form asks if every point in the escaping set of an arbitrary transcendental entire function can be joined to infinity by a curve in the escaping set.
This was answered in the negative by the 2011 paper of Rottenfusser, Rückert, Rempe, and Schleicher (RRRS) by constructing a tract that produces a function that cannot contain such a curve. In the same paper, it was also shown that if the function was of finite order, that is, log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )$ as $\left | z \right |\rightarrow \infty$, then every point in the escaping set can indeed be connected to infinity by a curve in the escaping set.
The counterexample $f$ used in the RRRS paper has growth such that log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )^{k}$ where $K > 12$ is an arbitrary constant. The question is, can this exponent, K, be decreased and can explicit calculations and counterexamples be performed and constructed that improve on this?[-]

The concept of a complex box mapping (or puzzle mapping) is a generalization of the classical notion of polynomial-like map to the case when one allows for countably many components in the domain and finitely many components in the range of the mapping. In one-dimensional dynamics, box mappings appear naturally as first return maps to certain nice sets, and hence one arrives at a notion of box renormalization. We say that a rational map is box renormalizable if the first return map to a well-chosen neighborhood of the set of critical points (intersecting the Julia set) has a structure of a box mapping. In our talk, we will discuss various features of general box mappings, as well as so-called dynamically natural box mappings, focusing on their rigidity properties. We will then show how these results can be used almost as 'black boxes' to conclude similar rigidity properties for box renormalizable rational maps. We will give several examples to illustrate this procedure, these examples include, most prominently, complex polynomials of arbitrary degree and their Newton maps. (The talk is based on joint work with Trevor Clark, Oleg Kozlovski, Dierk Schleicher and Sebastian van Strien.)[-]

There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). We will discuss several natural decompositions that arise in the study of rational maps, such as Pilgrim's canonical decomposition and Levy decomposition (by Bartholdi and Dudko). I will also introduce a new decomposition of rational maps based on the topology of their Julia sets (obtained jointly with Dima Dudko and Dierk Schleicher). At the end of the talk, we will briefly consider connections of this novel decomposition to geometric group theory and self-similar groups.[-]

We construct a transcendental entire function for which infinitely many Fatou components share the same boundary. This solves the long-standing open problem whether Lakes of Wada continua can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. Our theorem also provides the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a recent question of Boc Thaler. Using the same techniques, we give new counterexamples to a conjecture of Eremenko concerning curves in the escaping set of an entire function. This is joint work with Lasse Rempe and James Waterman.[-]

Combination theorems play an important role in several areas of dynamics, geometry, and group theory. In this talk, we will expound a framework to conformally combine Kleinian (reflection) groups and (anti-)holomorphic rational maps in a single dynamical plane. In the anti-holomorphic setting, such hybrid dynamical systems are generated by Schwarz reflection maps arising from univalent rational maps. A crucial technical ingredient of this study is a recently developed David surgery technique that turns hyperbolic conformal dynamical systems to parabolic ones. We will also mention numerous consequences of this theory, including 1. an explicit dynamical connection between various rational Julia and Kleinian limit sets,2. existence of new classes of welding homeomorphisms and conformally removable Julia/limit sets, and3. failure of topological orbit equivalence rigidity for Fuchsian groups acting on the circle.[-]

A transcendental entire function with bounded singular set that is hyperbolic and has a unique Fatou component is said to be of disjoint type. The Julia set of any disjoint-type function of finite order is known to be a collection of curves that escape to infinity and form a Cantor bouquet, i.e., a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush. We show that there exists $f$ of disjoint type whose Julia set $J(f)$ is a collection of escaping curves, but $J(f)$ is not a Cantor bouquet. On the other hand, we prove that if $f$ of disjoint type and $J(f)$ contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then $J(f)$ must be a Cantor bouquet. This is joint work with L. Rempe.[-]