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Nekhoroshev theory and discrete averaging

V. Gelfreich, A. Vieiro

Abstract

This paper contains a proof of the Nekhoroshev theorem for quasi-integrable symplectic maps. In contrast to the classical methods, our proof is based on the discrete averaging method and does not rely on transformations to normal forms. At the centre of our arguments lies the theorem on embedding of a near-the-identity symplectic map into an autonomous Hamiltonian flow with exponentially small error.

Nekhoroshev theory and discrete averaging

Abstract

This paper contains a proof of the Nekhoroshev theorem for quasi-integrable symplectic maps. In contrast to the classical methods, our proof is based on the discrete averaging method and does not rely on transformations to normal forms. At the centre of our arguments lies the theorem on embedding of a near-the-identity symplectic map into an autonomous Hamiltonian flow with exponentially small error.

Paper Structure

This paper contains 13 sections, 10 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Nekhoroshev theorem for a near-integrable map
  3. A priori bounds and strategy of the proof
  4. Interpolating vector fields
  5. Embedding a symplectic near-the-identity map into an autonomous Hamiltonian flow
  6. Interpolating flow near a fully resonant torus
  7. Exponentially accurate interpolation
  8. Long term stability of actions
  9. Nucleus of a resonance
  10. Final comments and conclusions
  11. Implicit function theorem
  12. Formal interpolation of a symplectic family
  13. Generating functions of exact symplectic maps

Key Result

Theorem 2.1

If a real-analytic exact symplectic map $F_\varepsilon$ satisfies the assumptions stated above and $h_0$ is strongly convex on a real neighbourhood of $B_R$, then there are positive constants $c_1,c_2,c_3$ such that for every initial condition $(I_0,\varphi_0)\in B_R\times \mathbb T^d$ one has

Theorems & Definitions (20)

  • Theorem 2.1: Nekhoroshev theorem
  • Theorem 2.2: long term stability of actions
  • Theorem 2.3: Dirichlet Cassels57
  • Lemma 2.4
  • proof
  • Lemma 3.1: a priori bounds
  • proof
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.2: formal interpolation
  • ...and 10 more