Iterated function systems that contract on average

UoM administered thesis: Phd

  • Authors:
  • Anthony Chiu

Abstract

Consider an iterated function system (IFS) that does not necessarily contract uniformly, but instead contracts on average after a finite number of iterations. Under some technical assumptions, previous work by Barnsley, Demko, Elton and Geronimo has shown that such an IFS has a unique invariant probability measure, whilst many (such as Peigné, Hennion and Hervé, Guivarc'h and le Page, Santos and Walkden) have shown that (for different function spaces) the transfer operator associated with the IFS is quasi-compact. A result due to Keller and Liverani allows one to deduce whether the transfer operator remains quasi-compact under suitable, small perturbations.The first part of this thesis proves a large deviations result for IFSs that contract on average using skew product transfer operators, a technique used by Broise to prove a similar result for dynamical systems.The remaining chapters introduce a notion of `coupled IFSs', an analogue of the traditional coupled map lattices where the base, unperturbed behaviour is determined by an underlying dynamical system. We use transfer operators and Keller and Liverani's theorem to prove that quasi-compactness of the transfer operator is preserved for `product IFSs' under small perturbations and for coupled IFSs. This allows us to prove a central limit theorem with a rate of convergence for the coupled IFS.

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Original languageEnglish
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Award date1 Aug 2015