2023/09/14 by Shaoyun Bai, Bai, Shaoyun, Guangbo Xu +1
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2309.07991
openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.