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

TL;DR

This paper proves that in compact toric symplectic manifolds, Hamiltonian diffeomorphisms with more fixed points than the Betti number have infinitely many simple periodic points, extending classical dichotomies and confirming a conjecture.

Contribution

It introduces the use of gauged linear sigma models and bulk deformations to analyze Hamiltonian dynamics on symplectic quotients, generalizing known results.

Findings

Hamiltonian diffeomorphisms with excess fixed points have infinitely many periodic points

Generalization of Franks' dichotomy to higher-dimensional toric manifolds

Confirmation of Hofer-Zehnder conjecture for toric manifolds

Abstract

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.