Masters Research - Masters of Philosophy (Mathematics)
Description
This thesis aims to provide an overview of the current works in the field of fractal sets produced via iterative mapping systems on circles. We will review the work of Frame, Mandelbrot and Neger(2005) in relation to circle inversion limit sets in the extended real plane. For the purpose of this thesis we will restrict our review to circle placements which are non-overlapping. Two circles will be classified non-overlapping if they have disjoint interiors and at most one point of contact between their boundaries. We will see how circle inversion limit sets can be created via an iterative system of non-expansive circle inversion mappings on non-overlapping circle placements, termed restricted iterated circle inversion systems (restricted ICISs). The work of Frame et al (2005) will be extended by developing a classification theorem for n non-overlapping circles in the extended real plane. This theorem predicts the structure of a given restricted ICIS limit set based on the placement of the initial inversion circles. We will see how the properties of circle inversion and circle inversion iteration can be extended to three dimension in the form of spherical inversion/ spherical inversion iteration. Once again we will limit ourselves to iterative systems using only non-expansive inversion mappings, termed restricted iterated spherical inversion systems (restricted ISISs). Limit sets formed under restricted ISISs on non-overlapping spherical placements will also be discussed. This thesis also explores iteration of mappings applied to circles in the extended complex plane. We will begin by reviewing the work of Mumford, Series and Wright (2002). Mumford et al (2002) formed fractal limit set by iterating systems of circle pairing Mobius transformations on specific non-overlapping circle placements. We will review the criteria needed for fractal limit sets to be formed under such an iterative system. Iteration of systems of complex circle inversion mappings will also be discussed as well as the structures of possible limit set. We will conclude this thesis with possible avenues for further work within the field of Fractal Prducing Iterative Mapping Systems on Circles.
Relation
University of Newcastle Research Higher Degree Thesis
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