The generic Conley conjecture for Hamiltonian diffeomorphisms

Let $(M,[?]$ be a closed symplectic manifold, and let a simple contractible periodic orbit mean a contractible periodic orbit that is not an iterate of a periodic orbit of lower period. The phrase “almost all” refers to the genericity notion for Hamiltonian diffeomorphisms used in the source.

Generic Conley conjecture. On every closed symplectic manifold, almost all Hamiltonian diffeomorphisms have infinitely many simple contractible periodic orbits.

This is a generic variant of the Conley conjecture, which asserts the same infinitude for every Hamiltonian diffeomorphism on almost all closed symplectic manifolds. The paper treats these assertions as open problems and proves the generic statement for very wide classes of symplectic manifolds.