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.[-]

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.[-]

Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques — adapting tools used tostudy mapping class groups — to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to.
This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.[-]

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.[-]