Hamiltonian elliptic dynamics on symplectic 4-manifolds

TL;DR

This paper investigates the prevalence of elliptic closed orbits in Hamiltonian flows on symplectic 4-manifolds, showing that such orbits are generically dense near non-Anosov energy surfaces.

Contribution

It demonstrates that for generic Hamiltonians, elliptic closed orbits are dense near non-Anosov energy surfaces, extending understanding of elliptic dynamics in 4D symplectic manifolds.

Findings

Elliptic closed orbits are dense near non-Anosov energy surfaces.

For generic Hamiltonians, elliptic orbits are prevalent.

Near any open set intersecting a non-Anosov energy surface, a Hamiltonian with an elliptic orbit exists.

Abstract

We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic.