Title:

Investigating the infinite spider's web in complex dynamics

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 explorethe 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 comple
ment 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 compo
nents having each of a number of orbit types, and we prove that
components with bounded orbits are quasiconformally homeomor
phic 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 J ulia 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 func
tion 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.
