The “two or infinity” conjecture of Hofer-Wysocki-Zehnder asserts that a Hamiltonian flow on a regular compact connected energy level has either two or infinitely many periodic orbits, when the energy level bounds a four-dimensional star-shaped domain.   I will explain the context for this conjecture, some key ideas behind recent joint work resolving it, and some related open questions that remain.  At the end of the talk, I will briefly discuss some aspects of a different work, going beyond periodic orbits to establish the dense existence of non-trivial invariant sets.