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).[-]

In complex dynamics it is usually important to understand the dynamical behavior of critical (or singular) orbits. For quadratic polynomials, this leads to the study of the Mandelbrot set and of its complement. In our talk we present a classification of some explicit families of the transcendental entire functions for which all singular values escape, i.e. functions belonging to the complement of the 'transcendental analogue' of the Mandelbrot set. This classification allows us to introduce higher dimensional analogues of parameter rays and to explore their properties. A key ingredient is a generalization of the famous Thurston's Topological Characterization of Rational Functions, but for the case of infinite rather than finite postsingular set. Analogously to Thurston's theorem, we consider the sigma-iteration on the Teichmüller space and investigate its convergence. Unlike the classical case, the underlying Teichmüller space is infinite-dimensional which leads to a completely different theory.[-]

A fundamental fact about Riemann surfaces is that they degenerate in a specific pattern — along thin annuli or wide rectangles. As it was demonstrated by W. Thurston in his realization theorem, we can often understand the dynamical system by establishing a priori bounds (a non-escaping property) in the near-degenerate regime. Similar ideas were proven to be successful in the Renormalization Theory of the Mandelbrot set. We will start the talk by discussing a dictionary between Thurston and Renormalization theories. Then we will proceed to neutral renormalization associated with the main cardioid of the Mandelbrot set. We will show how the Transcendental Dynamics naturally appear on the renormalization unstable manifolds. In conclusion, we will describe uniform a priori bounds for neutral renormalization — joint work with Misha Lyubich.[-]

In this talk we present bifurcation phenomena in natural families of rational or (transcendental) meromorphic functions of finite type $\left\{f_{\lambda}:=\varphi_{\lambda} \circ f_{\lambda_{0}} \circ \psi_{\lambda}^{-1}\right\}_{\lambda \in M}$, where $M$ is a complex connected manifold, $\lambda_{0} \in M, f_{\lambda_{0}}$ is a meromorphic map and $\varphi_{\lambda}$ and $\psi_{\lambda}$ are families of quasiconformal homeomorphisms depending holomorphically on $\lambda$ and with $\psi_{\lambda}(\infty)=\infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits the domain, while its multiplier tends to zero, proving a conjecture from [Fagella, Keen, 2019]. We also present the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $\mathcal{J}$-unstability, converging to a theorem in the spirit of Mañé-Sad-Sullivan's celebrated result.[-]

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.[-]

W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples:
--The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and
--The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold.
In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.[-]

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.[-]