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Cantor Minimal Systems and Dimension Groups
AuthorLuciano, Cyrus Anthony
AdvisorBlackadar, Bruce E.
Mathematics and Statistics
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The theory of general dynamical systems evolved originally in the context of modeling movement in physical systems. Consequently, traditional dynamics is viewed through the lens of differential and difference equations using notions such as state space, trajectory and attractors. Topological dynamics is a generalization and abstraction of these concepts in the context of topological spaces and homeomorphisms. Dimension groups provide a classification up to strong orbit equivalence of minimal Z^d-actions on a Cantor set. The range of such dimension groups for d>1 is still an open question. It appears as if symbolic dynamical systems may be a fruitful approach to this question. In this work, the relationship is explored between Cantor minimal systems, symbolic dynamical systems and dimension groups using properly ordered Bratteli diagrams and their associated Bratteli-Vershik systems. In order to illustrate this relationship we develop the examples of general odometer systems and irrational rotations on the Cantorized circle. Both the K_0 and K^0 groups are calculated using Bratteli diagrams and directed graphs, respectively. Furthermore, the substitution dynamical system associated to a specified irrational rotation is identified showing the method by which one may move between Cantor minimal systems and symbolic dynamical systems.
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