Quantum Steenrod powers and Hamiltonian maps

Abstract

We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, \omega)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff--Salamon's list, which predicts an obstruction to the existence of such special Hamiltonian diffeomorphisms in terms of genus zero numerical invariants. 2. If a Hamiltonian possesses a periodic orbit in a non-torsion homology class, then it has infinitely many simple periodic points. The same conclusion holds if the manifold is not geometrically uniruled and the diffeomorphism is minimal for rational Floer homology. These two general results complement the known cases of the Hofer--Zehnder conjecture. 3. If a Hamiltonian diffeomorphism possesses a symplectically degenerate maximum, then it has infinitely many simple periodic points. This resolves an open question which stems from the work of Ginzburg and G\"urel. We also establish new cases of the generic Conley conjecture: infinitely many periodic points for generic Hamiltonian diffeomorphisms. The proofs rely on a systematic application of the integral Hamiltonian Floer theory package developed by the first and fourth author, a K\"unneth isomorphism in equivariant Floer homology, and new quantitative analysis of quantum power maps, which is of independent interest.