Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity

TL;DR

This paper investigates Hamiltonian diffeomorphisms in Euclidean space, showing that under certain conditions, they must have simple periodic orbits of arbitrarily large period if they possess fixed points not essential from a homological viewpoint.

Contribution

It establishes the existence of arbitrarily large period simple periodic orbits for a class of Hamiltonian diffeomorphisms with specific fixed point properties.

Findings

Existence of arbitrarily large period simple periodic orbits.

Fixed points not necessary from a homological perspective imply periodic orbits.

Results apply to Hamiltonian systems linear and hyperbolic at infinity.

Abstract

We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have simple periodic orbits of arbitrarily large period when it has fixed points which are not necessary from a homological perspective.