On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms

TL;DR

This paper proves the existence of infinitely many non-contractible periodic orbits of arbitrarily large period for certain Hamiltonian diffeomorphisms on closed symplectic manifolds, under specific non-degeneracy and homological conditions.

Contribution

It establishes new conditions guaranteeing infinitely many non-contractible periodic orbits in Hamiltonian dynamics on atoroidal symplectic manifolds.

Findings

Existence of non-contractible periodic orbits of arbitrarily large period

Conditions under which these orbits are guaranteed

Finite set of one-periodic orbits in a given homology class

Abstract

We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non-contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a non-degenerate (or even isolated and homologically non-trivial) periodic orbit with non-zero homology class and the set of one-periodic orbits in that class is finite.