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Algebraic Structure of Dynamical Systems

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A dynamical system is a mathematical object which describes the motion of a set of points over time. Dynamical systems can be used to study differential equations, cryptography, computer science, and even biology. Viewed as a purely mathematical object, one can ask questions about the behavior of the dynamical system based on the structure of algebraic objects associated with it. In this project we study two algebraic objects, centralizers and topological full groups, associated to symbolic dynamical systems. The centralizer group tells us about the symmetries a system possesses. Results relating to the centralizer historically have indicated that the more complex the dynamical system is, captured by the Topological Entropy, the more structure its centralizer has. Similarly, low complexity systems have been shown to have very simple centralizers. This seems to suggest that one can recover information about the dynamical system based upon its centralizer group. In particular, if a system is known to have a certain centralizer group, we might want to draw conclusions about the complexity of the system. In this project we present a class of high complexity systems which have a very rigid centralizer, which shows the relationship is more subtle than may have been originally thought. We also study the topological full group of a dynamical system. This group completely defines the system up to time reversal. We apply numerical estimates to draw conclusions about the algebraic properties of this group. In particular, we seek to know when the topological full group of a dynamical system is amenable. Amenability is an algebraic property that can bethought of as having a probability measure on G. This measure would answer the question given a subset A of G, what is the probability that a random element of G is in A We apply Grigorchuks amenability criterion to answer this question.

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  • Theoretical Mathematics

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