On the Hofer-Zehnder conjecture on weighted projective spaces

TL;DR

This paper extends Shelukhin's homology-based Hofer-Zehnder conjecture proof to weighted projective spaces, showing that certain Hamiltonian diffeomorphisms with many fixed points have infinitely many periodic points.

Contribution

It generalizes the Hofer-Zehnder conjecture to symplectic orbifolds, specifically weighted projective spaces, and establishes conditions for infinite periodic points.

Findings

Proves the conjecture for weighted projective spaces as symplectic orbifolds.

Shows that fixed points with multiplicity imply infinitely many periodic points.

Extends homology methods to orbifold settings.

Abstract

We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.