Thompson-Like Groups for Quadratic Rational Julia Sets

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Senior Projects Spring 2013

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Weinrich-Burd, Jasper O.
Bard Digital Commons
Thompson's groups; Julia set; geometric group theory; dynamical systems; complex dynamics; Algebra; Dynamical Systems; Geometry and Topology
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Julia sets are fractals that arise in the study of dynamical systems on the complex plane. Recently, Belk and Forrest investigated a group of "piecewise-linear" homeomorphisms on the Julia set for the function f(z) = z2-1. This group closely resembles Thompson's group T, a finitely generated group of piecewise-linear functions on the unit circle. Inspired by Belk and Forrest's work, we examine the Julia set for the function f(z)=z-2-1, which we refer to as the Bubble Bath. Because f is rational, the external angles used by Belk and Forrest are not available. Instead our approach makes heavy use of symbolic dynamics. In particular, we show how to assign an address to every point in the Bubble Bath. We use these addresses to define a group TBB. We prove that TBB is generated by four elements, that it contains T, and that it is a semi-direct product of its double commutator subgroup with S3. We also prove that its double commutator subgroup is an infinite simple group. Finally, we briefly investigate homeomorphisms of certain other rational Julia sets.