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Minimal multiplicity of prime closed orbits without additional assumptions

Determine whether, without any extra geometric or dynamical conditions on M, a Reeb flow on the boundary of a smooth star-shaped domain in R^{2n} must have more than one prime closed orbit, or more than two prime closed orbits if the flow is non-degenerate.

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Background

Beyond convexity, dynamical convexity, or symmetry, very little is known about multiplicity of prime closed Reeb orbits in the general setting of star-shaped hypersurfaces. The authors highlight that even the existence of more than one (or more than two non-degenerate) prime closed orbit is not established in full generality.

Their main theorems provide lower bounds under dynamical convexity, but the minimal multiplicity question in the unrestricted case remains unresolved.

References

For instance, without extra conditions on M, it is not even known that there must be more than one prime orbit or more than two if the flow is non-degenerate.

— Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures (2410.13093 - Cineli et al., 16 Oct 2024) in Section 1.1 (Introduction)