A hierarchy of chaotic topological dynamics

We explore the topological dynamics concepts of transitivity, total transitivity, and mixing on the interval, circle, torus, and sphere. Transitivity, total transitivity, and mixing form a hierarchy and we investigate, for each space, at which point in the hierarchy chaos becomes necessary. In addition to a novel proof that total transitivity and mixing are equivalent on the interval, we show that mixing implies chaos for functions on the circle and for toral automorphisms. This thesis is in partial completion of honors in mathematics at Davidson College.
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Abstract/Description: We explore the topological dynamics concepts of transitivity, total transitivity, and mixing on the interval, circle, torus, and sphere. Transitivity, total transitivity, and mixing form a hierarchy and we investigate, for each space, at which point in the hierarchy chaos becomes necessary. In addition to a novel proof that total transitivity and mixing are equivalent on the interval, we show that mixing implies chaos for functions on the circle and for toral automorphisms. This thesis is in partial completion of honors in mathematics at Davidson College.
Subject(s): Chaotic behavior in systems. -- Topological dynamics.
Date Issued: 2013