On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics

TL;DR

This paper proves the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics on broad classes of symplectic manifolds, addressing an open problem in the field.

Contribution

It establishes the conjecture for non-contractible periodic orbits, expanding understanding of periodic orbit multiplicity in Hamiltonian systems.

Findings

Proves the Hofer-Zehnder conjecture for non-contractible orbits.

Shows that non-contractible orbits are homologically unnecessary.

Applies to wide classes of symplectic manifolds.

Abstract

In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds. Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphisms has infinitely many periodic orbits if it has "homologically unnecessary periodic orbits"". For example, non-contractible periodic orbits are homologically unnecessary periodic orbits because Floer homology of non-contractible periodic orbits is trivial. We prove Hofer-Zehnder conjecture for non-contractible periodic orbits for very wide classes of symplectic manifolds.