Thompson-Like Groups for Quadratic Rational Julia Sets

Citation data:

Senior Projects Spring 2013

Publication Year:
2013
Usage 26
Abstract Views 20
Downloads 6
Repository URL:
http://digitalcommons.bard.edu/senproj_s2013/149; http://digitalcommons.bard.edu/cgi/viewcontent.cgi?article=1160&context=senproj_s2013; https://digitalcommons.bard.edu/cgi/viewcontent.cgi?article=1160&context=senproj_s2013
Author(s):
Weinrich-Burd, Jasper O.
Publisher(s):
Bard Digital Commons
Tags:
Thompson's groups; Julia set; geometric group theory; dynamical systems; complex dynamics; Algebra; Dynamical Systems; Geometry and Topology
artifact description
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.