Simple Smale flows on $S^3$ using twisted 4-band templates
Keywords:
Smale flows, Templates, Knots, Attractors, RepellersAbstract
We investigate the linking structures of attractor-repeller pairs in simple Smale flows on $S^3$ in which the chaotic saddle sets are modeled by four-band templates with twisted bands. We obtain 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 $a \cup r$ in a simple Smale flow with chaotic saddle set modeled by embedded template $U^+$ is either a Hopf link or a trefoil and meridian. We modify template $U^+$ by placing half-twists in selected bands. For simple Smale flows on $S^3$ with chaotic saddle sets modeled by these modified templates, we find that such simple Smale flows are realizable and that $a \cup r$ must be a Hopf link, a figure-8 knot and meridian, a trefoil and meridian, or a cinquefoil and meridian.
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