On the Hofer-Zehnder conjecture for semipositive symplectic manifolds

TL;DR

This paper proves that on certain semipositive symplectic manifolds, Hamiltonian diffeomorphisms with sufficiently many fixed points necessarily have infinitely many periodic points, extending Shelukhin's result on the Hofer-Zehnder conjecture.

Contribution

It generalizes Shelukhin's Hofer-Zehnder conjecture result to the semipositive symplectic setting with semisimple quantum homology.

Findings

Hamiltonian diffeomorphisms with many fixed points have infinitely many periodic points

The result applies to closed semipositive symplectic manifolds with semisimple quantum homology

Generalizes previous results to a broader class of symplectic manifolds

Abstract

We show that, on a closed semipositive symplectic manifold with semisimple quantum homology, any Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the manifold, must have infinitely many periodic points. This generalizes to the semipositive setting the beautiful result of Shelukhin on the Hofer-Zehnder conjecture.