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Bumpy metric theorem in the sense of Man{é} for non-convex Hamiltonians

Shahriar Aslani, Patrick Bernard

TL;DR

This work extends Mañé genericity to non-convex Hamiltonians by introducing fiberwise isoenergetic non-degeneracy and neat times, and by proving that admissible potential perturbations can weakly perturb the restricted linearized return map into a desired symplectic class. A fibered normal form around isoenergetically non-degenerate orbit segments provides a controlled framework for perturbations, reducing the problem to a bilinear control system and establishing a surjectivity result for the controlled linearized flow. Harnessing parametric transversality, the authors show that for generic potentials the zero-energy level is regular and all zero-energy periodic orbits are non-degenerate with their restricted Poincaré maps avoiding any fixed conjugacy-invariant subset of Sp(2d) with empty interior. They further prove that orbits confined to a submanifold transverse to the vertical are also generically avoided, via Sard-type arguments and finite-dimensional perturbations, yielding a robust non-degeneracy/regularity picture for non-convex Hamiltonians in the Mañé sense.

Abstract

We prove a bumpy metric theorem in the sense of Mañe for non-convex Hamiltonians that are satisfying a certain geometric property.

Bumpy metric theorem in the sense of Man{é} for non-convex Hamiltonians

TL;DR

This work extends Mañé genericity to non-convex Hamiltonians by introducing fiberwise isoenergetic non-degeneracy and neat times, and by proving that admissible potential perturbations can weakly perturb the restricted linearized return map into a desired symplectic class. A fibered normal form around isoenergetically non-degenerate orbit segments provides a controlled framework for perturbations, reducing the problem to a bilinear control system and establishing a surjectivity result for the controlled linearized flow. Harnessing parametric transversality, the authors show that for generic potentials the zero-energy level is regular and all zero-energy periodic orbits are non-degenerate with their restricted Poincaré maps avoiding any fixed conjugacy-invariant subset of Sp(2d) with empty interior. They further prove that orbits confined to a submanifold transverse to the vertical are also generically avoided, via Sard-type arguments and finite-dimensional perturbations, yielding a robust non-degeneracy/regularity picture for non-convex Hamiltonians in the Mañé sense.

Abstract

We prove a bumpy metric theorem in the sense of Mañe for non-convex Hamiltonians that are satisfying a certain geometric property.
Paper Structure (6 sections, 16 theorems, 83 equations)

This paper contains 6 sections, 16 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Normal form near orbit segments of non-convex Hamiltonian systems
  4. Perturbing the linearized maps.
  5. Parametric transversality
  6. Orbits inside a submanifold.

Key Result

Theorem 1

Consider a smooth Hamiltonian $H(q,p):T^*M\to \mathbb{R}$, and a periodic orbit $\theta(t)$ of the Hamiltonian vector field of $H$. Assume that $\theta$ admits a neat time $t_0 \in \mathbb{R}$ such that $H$ is fiberwise isoenergetically non-degenerate at $\theta(t_0)$. Then the map is weakly open, meaning that the image of each non-empty open set contains a non-empty open set.

Theorems & Definitions (28)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Definition 2
  • proof : Proof of (1)
  • proof : Proof of (2)
  • ...and 18 more