A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gürel, I will present a "forced existence" result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension:
If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits.
If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption.
These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gürel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.
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