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Dynamics of Equicontinuous Group Actions on Cantor Sets

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Title: Dynamics of Equicontinuous Group Actions on Cantor Sets
Author(s): Dyer, Jessica C.
Advisor(s): Hurder, Steven
Contributor(s): Lukina, Olga; Furman, Alexander; Rosendal, Christian; Ugarcovici, Ilie
Department / Program: Mathematics, Statistics, and Computer Science
Graduate Major: Mathematics
Degree Granting Institution: University of Illinois at Chicago
Degree: PhD, Doctor of Philosophy
Genre: Doctoral
Subject(s): dynamics group actions Cantor sets equicontinuous minimal group chain inverse limit
Abstract: In this thesis, we consider the class of minimal equicontinuous Cantor dynamical systems. We show that every such system can be represented by a group chain inverse limit system, and conversely that every group chain yeilds a minimal equicontinuous Cantor dynamical system. This gives us a concrete representation of minimal equicontinuous Cantor dynamical systems, which makes them easier to work with. We use this representation to classify such systems as regular, weakly regular, or irregular, extending work by Fokkink and Oversteegen. We show that such systems can be equivalently classified as regular, weakly regular, or irregular according to the number of orbits of the action by the Autormorphism group, or equivalently according to the number of equivalence classes of group chains associated to the system. We give examples of group chains of each level of regularity. We introduce a new invariant of such systems, called the discriminant group, and show that its cardinality is related to the classification of the system as regular, weakly regular, or irregular.
Issue Date: 2016-02-16
Genre: thesis
URI: http://hdl.handle.net/10027/20185
Rights Information: Copyright 2015 Jessica C. Dyer
Date Available in INDIGO: 2016-02-16
Date Deposited: 2015-12
 

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