  We prove that for any compact toric symplectic manifold, if a Hamiltonian
diffeomorphism admits more fixed points, counted homologically, than the total
Betti number, then it has infinitely many simple periodic points. This provides
a vast generalization of Franks' famous two or infinity dichotomy for periodic
orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a
conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key
novelty is the application of gauged linear sigma model and its bulk
deformations to the study of Hamiltonian dynamics of symplectic quotients.
