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We prove that for any compact toric symplectic manifold, if a Hamiltoniandiffeomorphism admits more fixed points, counted homologically, than the totalBetti number, then it has infinitely many simple periodic points. This providesa vast generalization of Franks' famous two or infinity dichotomy for periodicorbits of area-preserving diffeomorphisms on the two-sphere, and establishes aconjecture attributed to Hofer-Zehnder in the case of toric manifolds. The keynovelty is the application of gauged linear sigma model and its bulkdeformations to the study of Hamiltonian dynamics of symplectic quotients.

Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model