Lorenz knots are prime. (English) Zbl 0595.58037

Williams, R. F.

Ergodic Theory Dyn. Syst. 4, 147-163 (1984).

Summary: Lorenz knots are the periodic orbits of a certain geometrically defined differential equation in \({\mathbb{R}}^ 3\). This is called the ”geometric Lorenz attractor” as it is only conjecturally the real Lorenz attractor. These knots have been studied by the author and Joan Birman via a ”knot-holder”, i.e. a certain branched two-manifold H. To show such knots are prime we suppose the contrary which implies the existence of a splitting sphere, \(S^ 2\). The technique of the proof is to study the intersection \(S^ 2\cap H\). A novelty here is that \(S^ 2\cap H\) is likewise branched.

MSC:

37G99

Local and nonlocal bifurcation theory for dynamical systems

Keywords:

Lorenz knots; periodic orbits; differential equation; geometric Lorenz attractor

References:

[1]	DOI: 10.1007/BF02684770 · Zbl 0484.58021 · doi:10.1007/BF02684770
[2]	DOI: 10.1007/BF02684769 · Zbl 0436.58018 · doi:10.1007/BF02684769
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[5]	Guckenheimer, The Hopf Bifurcation (1976) · Zbl 0443.58017