Date of Award
12-1-2024
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Sullivan, Michael
Abstract
We study the linking structure of the attractor-repeller pairs in simple Smale flows on the 3-sphere in which the chaotic saddle set is modeled by four-band templates with twisted bands. This is a small step in an attempt to classify simple Smale flows on S³. We obtain three new theorems which illustrate that the dynamics of simple Smale flows are sensitive to half-twists in the bands of the embedded template. Haynes and Sullivan showed that the attractor-repeller pair a∪r in a simple Smale flow with chaotic saddle set modeled by embedded template U⁺ is either a Hopf link or a trefoil and meridian [19]. By placing a single half-twist in a selected band of U⁺, we obtain four new templates that model chaotic saddle sets. For simple Smale flows on S³ with chaotic saddle sets modeled by those templates, we find that such simple Smale flows are realizable and that a∪r must be a Hopf link, a figure-8 knot and meridian, a trefoil and meridian, or a cinquefoil and meridian.
Access
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