On the intersections of projected Hamiltonian orbits in cotangent bundles

TL;DR

This paper investigates the behavior of Hamiltonian trajectories in cotangent bundles, showing that generic projected orbits typically have discrete intersections, and these intersections can often be eliminated through perturbations in higher-dimensional bases.

Contribution

It establishes generic intersection properties of projected Hamiltonian orbits and demonstrates how to perturb away intersections in higher-dimensional settings, including Reeb and Finsler flows.

Findings

Projected trajectories have discrete intersections for generic level sets.

Intersections can be perturbed away in bases of dimension three or more.

Results apply to Reeb flows and non-reversible Finsler flows.

Abstract

We study the generic behavior of Hamiltonian trajectories on a regular level set in the cotangent bundle, after projection to the base. We prove that for a generic submersive level set, projected trajectories have discrete (self-)intersections. Additionally, fixing end-point fibers, we prove that all intersections can be perturbed away if the base has dimension at least three. In particular, this applies to periodic orbits, and both results hold for Reeb flows on fiber-wise star-shaped hypersurfaces, including non-reversible Finsler flows, which answers a question of Rademacher. In the proof we make use of a multi-jet transversality theorem.