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On Non-KAM Invariant Circles for Area-Preserving Twist Maps

Jiashen Guo, Yi Liu, Lin Wang

Abstract

In this note, we investigate the dynamics of invariant circles in area-preserving twist maps. The invariant circles under consideration lie beyond the applicability of classical KAM theory, as the perturbations involved exceed the scope of standard KAM methods. By integrating both constructive and non-constructive techniques, we establish several results on the existence and breakdown of such non-KAM invariant circles. These findings provide new evidence in support of affirmative answers to two open questions posed by Mather in 1998.

On Non-KAM Invariant Circles for Area-Preserving Twist Maps

Abstract

In this note, we investigate the dynamics of invariant circles in area-preserving twist maps. The invariant circles under consideration lie beyond the applicability of classical KAM theory, as the perturbations involved exceed the scope of standard KAM methods. By integrating both constructive and non-constructive techniques, we establish several results on the existence and breakdown of such non-KAM invariant circles. These findings provide new evidence in support of affirmative answers to two open questions posed by Mather in 1998.
Paper Structure (14 sections, 17 theorems, 100 equations, 1 figure)

This paper contains 14 sections, 17 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic Setting
  3. Related work by Herman
  4. On Questions (I) and (II)
  5. On Questions (III) and (IV)
  6. A Positive Evidence for Question 1
  7. Partial Support for Question 2
  8. More on the Set $U_\delta^{3-\varepsilon} \setminus U_\delta^3$
  9. Preliminaries
  10. The Herman Formula
  11. The Arnold Family
  12. Proof of Theorem \ref{['M0']}
  13. Proof of Theorem \ref{['M01']}
  14. Proof of Theorem \ref{['M1']}

Key Result

Proposition 1.4

Figures (1)

  • Figure 1: Arnold tongues: level sets $T_\rho$ of the rotation number $\rho$ for $w,\sigma\in[0,1]$. Tongues of every rational number are connected and have nonempty interior. Besides, if $\theta$ is an irrational number, $T_\theta$ may not have an interior point.

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 1
  • Remark 1.6
  • Definition 1.7
  • Theorem 2
  • Remark 1.8
  • ...and 21 more