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One distinguishing feature of Hamiltonian systems is that such systems tend to have innitely many periodic orbits. There are, however, simple exceptions such as the standard two-sphere: an irrational rotation of the two-sphere has only two periodic orbits and they are both xed points { the poles. Yet a celebrated theorem of Franks asserts that once a Hamiltonian dieomorphism of the two-sphere has more than two xed points it has innitely many periodic orbits, and one conjecture inspired by Franks' theorem is that a Hamiltonian dieomorphism with xed points that are unnecessary from a homological or geometrical perspective must have innitely many periodic orbits. In this talk we will discuss some higher-dimensional results supporting this con- jecture and, if time permits, show that in certain instances the presence of one hyperbolic xed point forces the existence of innitely many periodic orbits.