Investigating the infinite spider's web in complex dynamics

Osborne, John William (2012). Investigating the infinite spider's web in complex dynamics. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000d4fd

Abstract

This thesis contains a number of new results on the topological and geometric properties of certain invariant sets in the dynamics of entire functions, inspired by recent work of Rippon and Stallard.

First, we explore the intricate structure of the spider's web fast escaping sets associated with certain transcendental entire functions. Our results are expressed in terms of the components of the complement of the set (the 'holes' in the web). We describe the topology of such components and give a characterisation of their possible orbits under iteration. We show that there are uncountably many components having each of a number of orbit types, and we prove that components with bounded orbits are quasiconformally homeomorphic to components of the filled Julia set of a polynomial. We prove that there are singleton periodic components and that these are dense in the Julia set.

Next, we investigate the connectedness properties of the set of points K( f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K( f) is to- tally disconnected if and only if each component of K (f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.

Finally we show that, if the Julia set of a transcendental entire function is locally connected, then it must take the form of a spider's web. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.