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Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity

Alberto Enciso, Manuel Garzón, Daniel Peralta-Salas

TL;DR

The paper studies generic, low-energy dynamics of natural Hamiltonians on the 2-torus, proving non-ergodicity and the coexistence of quasi-periodic invariant tori with hyperbolic periodic orbits on small energy levels. It combines a detailed Birkhoff normal form near a nondegenerate minimum with a rescaled Poincaré-map analysis to establish persistence of invariant tori via Moser's twist theorem and to detect resonances via a subharmonic Melnikov method. A careful, explicit control of the reversed-action function $J$ and its $\,\varepsilon$-expansion, together with structured Melnikov-type coefficients, yields an open dense set of potentials for which arbitrarily many hyperbolic periodic orbits appear on each low-energy level. Overall, the work provides a rigorous, quantitative picture of non-ergodicity and intricate hyperbolic structure in generic low-energy dynamics of natural Hamiltonians with two degrees of freedom.

Abstract

We prove that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential on $\mathbf{T}^2$ exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. Hence, quasi-periodic motion and hyperbolic behavior typically coexist in the low-energy dynamics of natural Hamiltonian systems with two degrees of freedom.

Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity

TL;DR

The paper studies generic, low-energy dynamics of natural Hamiltonians on the 2-torus, proving non-ergodicity and the coexistence of quasi-periodic invariant tori with hyperbolic periodic orbits on small energy levels. It combines a detailed Birkhoff normal form near a nondegenerate minimum with a rescaled Poincaré-map analysis to establish persistence of invariant tori via Moser's twist theorem and to detect resonances via a subharmonic Melnikov method. A careful, explicit control of the reversed-action function and its -expansion, together with structured Melnikov-type coefficients, yields an open dense set of potentials for which arbitrarily many hyperbolic periodic orbits appear on each low-energy level. Overall, the work provides a rigorous, quantitative picture of non-ergodicity and intricate hyperbolic structure in generic low-energy dynamics of natural Hamiltonians with two degrees of freedom.

Abstract

We prove that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential on exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. Hence, quasi-periodic motion and hyperbolic behavior typically coexist in the low-energy dynamics of natural Hamiltonian systems with two degrees of freedom.

Paper Structure

This paper contains 11 sections, 16 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Computation of the Birkhoff normal form at the minimum
  3. Computation of the Poincaré map
  4. Proof of Theorem \ref{['MainTheorem']}: non-ergodicity
  5. Proof of Theorem \ref{['MainTheorem']}: hyperbolic periodic orbits
  6. Structure of the reversed action function
  7. Auxiliary lemmas
  8. Proof of Proposition \ref{['J-Proposition']}
  9. Proof of Proposition \ref{['Pro-Hyp0']}
  10. Proof of Proposition \ref{['Pro-Hyp1']}
  11. The class $\mathcal{V}_{N_0}$ is dense in $C^\infty(\mathbb T^2)$

Key Result

Theorem 1.1

Let $N_0$ be a positive integer. There exists a generic set $\mathcal{V}\subset C^\infty(\mathbb T^2)$ and a small constant $\delta_0$ such that the Hamiltonian system $H_V$ with $V\in\mathcal{V}$ exhibits a positive measure set of invariant tori and at least $N_0$ hyperbolic periodic orbits on each

Theorems & Definitions (23)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Corollary 2.3
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Theorem 4.1
  • Theorem 5.1
  • ...and 13 more