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On the intersections of projected Hamiltonian orbits in cotangent bundles

Lucas Dahinden, Jacobus de Pooter

TL;DR

This work generalizes Rademacher's results on generic non-intersections of geodesics to broad Hamiltonian dynamics on cotangent bundles by focusing on the projected base trajectories of regular energy level sets. The authors introduce a robust perturbation framework using Hamiltonians with $\Sigma$ regular and $\pi_Q$ a submersion, and they prove that generic Hamiltonians exhibit discrete base-intersections and no self-intersections for nontrivial trajectories, with stronger results in dimensions $\dim Q \ge 3$. The core method employs multijet transversality in jet spaces of submanifolds, translating surface perturbations into perturbations of flows via radial/contact models. The results apply to Reeb flows on fiberwise star-shaped hypersurfaces and even non-reversible Finsler flows, providing new answers to questions on geometrically distinct orbits and chord growth, and yielding explicit transversality-based perturbation schemes. Overall, the paper advances the understanding of generic intersection properties in Hamiltonian dynamics and offers a systematic, jet-theoretic route to discrete-intersection results in high-dimensional settings.

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.

On the intersections of projected Hamiltonian orbits in cotangent bundles

TL;DR

This work generalizes Rademacher's results on generic non-intersections of geodesics to broad Hamiltonian dynamics on cotangent bundles by focusing on the projected base trajectories of regular energy level sets. The authors introduce a robust perturbation framework using Hamiltonians with regular and a submersion, and they prove that generic Hamiltonians exhibit discrete base-intersections and no self-intersections for nontrivial trajectories, with stronger results in dimensions . The core method employs multijet transversality in jet spaces of submanifolds, translating surface perturbations into perturbations of flows via radial/contact models. The results apply to Reeb flows on fiberwise star-shaped hypersurfaces and even non-reversible Finsler flows, providing new answers to questions on geometrically distinct orbits and chord growth, and yielding explicit transversality-based perturbation schemes. Overall, the paper advances the understanding of generic intersection properties in Hamiltonian dynamics and offers a systematic, jet-theoretic route to discrete-intersection results in high-dimensional settings.

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.
Paper Structure (30 sections, 23 theorems, 70 equations, 1 figure)

This paper contains 30 sections, 23 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and Background
  3. Generalization
  4. Notable special cases
  5. Many geometrically distinct geodesics
  6. Proofs
  7. (Step 1)
  8. (Step 2)
  9. (Step 3)
  10. (Conclusion)
  11. Sketch of proof of Theorem \ref{['thm:discreteIntersections']}
  12. Preliminaries
  13. Jet bundles
  14. Jets of submanifolds
  15. Bundle of submanifolds
  16. ...and 15 more sections

Key Result

Theorem 1

Let $M$ be a closed manifold of dimension $\geq 3$ and let $p,q\in M$ be two points, not necessarily distinct. For a $C^r$-generic set of Riemannian metrics ($r\geq 2$), geodesic segments connecting $p$ to $q$ have no self-intersection, and geometrically distinct segments have no interior intersecti

Figures (1)

  • Figure 1: On the left hand side we see a fiber $\Sigma_q$ with six marked points: The inflection points and their parallels in green and red, respectively. On the right hand side we see the torus $\Sigma_q\times\Sigma_q$ with the coordinate lines of the red and green points. The blue lines are the fiber ${\mathcal{H}}^1_{q,q}$. Note that it consists of two connected components: An off-diagonal circle representing the antipodals which curves at the inflection points times their parallels, and a contractible circle united with the diagonal representing the isopodals.

Theorems & Definitions (60)

  • Theorem : Rademacher
  • Definition 1.2
  • Remark 1.3
  • Theorem 1
  • Definition 1.4
  • Example 1.5
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Remark 1.6
  • ...and 50 more