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Periodic orbits and fixed points of a \(C^ 1\) orientation-preserving embedding of \(D^ 2\). (English) Zbl 0711.57019
Summary: We prove that any periodic orbit \({\mathcal O}\) of a \(C^ 1\) orientation- preserving embedding f of the 2-disc \(D^ 2\) is linked with a fixed point \({\mathcal P}\) in the sense that the corresponding periodic orbits \(\{\Phi_ t({\mathcal O})\}_{t\geq 0}\) and \(\{\Phi_ t({\mathcal P})\}_{t\geq 0}\) of any torus flow \(\Phi_ t\) suspending f are linked as knots in \(S^ 3\).

MSC:
57R40 Embeddings in differential topology
37G99 Local and nonlocal bifurcation theory for dynamical systems
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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[1] DOI: 10.1090/S0273-0979-1988-15685-6 · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
[2] DOI: 10.1007/BFb0061407 · doi:10.1007/BFb0061407
[3] Gambaudo, Ann. Inst. H. Poincar? Phys. Th?or
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