Abstracts

Martí-Pete: On the dimension of the boundaries of attracting basins of entire maps 

Bogdanov - Infinite dimensional Thurston theory

Thurston's iteration is well-defined in many relevant cases when the set of marked points is infinite, e.g., for entire functions with escaping singular orbits (or, more generally, staying sufficiently close to infinity). As in the ``finite'' case, the fixed point of the iteration corresponds to an entire function with the post-singular behaviour modeled by the Thurston map. However, the underlying Teichmüller space is infinite-dimensional which requires a different approach compared to the finite-dimensional case.

Starting with polynomials and moving towards essentially more general non-explicitly given families of transcendental entire functions, we overview the principal ideas allowing to deduce the convergence of the iteration for a generic mode of escape.

Mukundan - tbc

Münch - Expanding rational expanding Thurston maps to R^3

Uniformly quasi-regular maps are a nice class to study if one aims to extend results of complex dimensions in the plane to real higher dimensions. However uniform quasi-regularity is a strong assumption and it is not trivial to find examples. Our construction provides maps such that each point can be iterated finitely many times without leaving the domain and for these iterates the constant is uniform. We will explain how to extend rational expanding Thurston maps f defined on the Riemann sphere S^2 to a mapping F that is defined on a set A containing S^2 in its interior to a map that is uniformly quasi-regular for all admissible iterates.

In the second part of the talk we will treat an application of this construction that is motivated by Sullivan’s dictionary. Sullivan’s dictionary provides an analogy between objects, conjectures and theorems in complex dynamics and the study of Kleinian groups. The aim of the talk is to explain a construction of a space-filling f-invariant curve that arises as the boundary of an immersed and severely folded plane. The analogous object from the Kleinian groups side is the Cannon-Thurston map, that is a Group-invariant Peano curve. 

Glynn -  Boundary stratifications of Hurwitz spaces

A Hurwitz space H parametrises holomorphic maps of Riemann surfaces (with certain points marked) that satisfy prescribed branching conditions. Abramovich, Corti and Vistoli describe a compactification of H that has a natural boundary stratification. For a Hurwitz space H parametrising maps to P^1, we show that the strata are in bijection with combinatorial objects called decorated trees (up to a suitable equivalence). We discuss applications to complex dynamics, and in particular to Thurston maps. 

Prochorov - Thurston theory for general finite type maps 

General finite-type maps, introduced by A. Epstein, capture many key transcendental dynamical systems (for example, those coming from entire or meromorphic functions with finitely many singular values) and also arise naturally in polynomial and rational dynamics. More precisely, we look at partially defined dynamical systems given by maps f: W -> X, where X is a compact Riemann surface, W is an open subset of X, and f has finitely many singular values (i.e., points where not all inverse branches of f are univalent). The goal of this talk is to extend Thurston theory to this setting and show that it is possible to derive Thurston-like characterization results in this very broad framework. This is work in progress. 

Schleicher - tbc

Dudko - The near-degenerate regime in transcendental dynamics

We consider transcendental maps of finite order, without asymptotic values, and with a controlled (of uniform width) structure of the line complex. Employing the Near-Degenerate Regime, we establish certain a priori bounds to quasiconformally (and equivariantly) identify transcendental and polynomial postcritical sets of Yoccoz maps (no neutral and no infinitely renormalizable dynamics) so thatthe dilatation depends only on the combinatorics but not on the degrees. Using dynamically meaningful approximations by polynomials, various realization and rigidity results are elevated from Polynomial Dynamics to the above-specified Transcendental Yoccoz setting. For postcritically finite maps, the results yield a Transcendental Thurston Realization Theorem. Joint work with Kostya Drach.