n-or-∞ conjecture for Reeb flows on star-shaped hypersurfaces

Establish the n-or-∞ conjecture for Reeb flows on the boundary M of a smooth star-shaped domain in R^{2n}: (i) determine whether the Reeb flow on M has at least n prime closed orbits (Multiplicity), and (ii) determine whether the Reeb flow on M has exactly n prime closed orbits whenever the flow is a Reeb pseudo-rotation, i.e., the number of prime closed orbits is finite (contact Hofer–Zehnder conjecture).

Background

The paper studies multiplicity and upper bounds for prime closed Reeb orbits on boundaries of star-shaped domains in R^{2n}. The authors formulate a comprehensive conjecture (the n-or-∞ conjecture) capturing expected optimal multiplicity and an upper bound when only finitely many prime closed orbits exist (Reeb pseudo-rotation). They prove sharp lower bounds under dynamical convexity and verify a particular case of the upper bound under symmetry and non-degeneracy, but the full conjecture in general remains open.

Part (C-M) is a classical multiplicity question, previously known under convexity/dynamical convexity assumptions. Part (C-HZ) is a contact analogue of the Hofer–Zehnder conjecture in Hamiltonian dynamics. The paper’s results advance these in special cases but do not resolve the general conjecture.