Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.582809
Title: Investigating the infinite spider's web in complex dynamics
Author: Osborne, John
Awarding Body: Open University
Current Institution: Open University
Date of Award: 2012
Availability of Full Text:
Access through EThOS:
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 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.582809  DOI: Not available
Share: