Speaker
Description
A transcendental entire function is said to be of finite type if it has only finitely many critical values (images of critical points) and asymptotic values (non-algebraic singularities of inverse branches). Functions of finite type are of significant interest in complex analysis and complex dynamics.
Eremenko and Lyubich showed that the class S of finite-type entire functions is stratified by finite-dimensional complex parameter space, consisting of functions all of which have the same topological mapping behaviour. For example, one such parameter space is given by pre- and post-compositions of the sine function with affine maps.
It is a natural question what can happen as functions in a given parameter space degenerate. For example, what are the possible limits of a sequence of functions in such a parameter space that leaves any compact subset thereof? In joint work with Prochorov, we show that this limiting behaviour can be extremely complicated. In fact, we show that, for any given finite set S, there exists a universal function for this set: A transcendental entire function f such that the possible pre-compositions of f with affine maps accumulate on all entire functions with the same singular values. In particular, the parameter space of such a universal function f accumulates on all
Moreover, using Bishop's technique of quasiconformal folding, we show that the pre-compositions in question accumulate even on every (not necessarily entire) finite-type map defined on a simply-connected domain and having singular set S. Given time, I will discuss consequences of this result for transcendental dynamics, in particular for the important problem of extending Thurston's characterisation of post-critically finite rational maps to the setting of transcendental dynamics.