Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Herman's converse KAM mechanism revisited

Yi Liu, Lin Wang

TL;DR

This work revisits Herman's converse KAM mechanism in the setting of area-preserving twist maps, showing that a bump perturbation is essential when employing more natural hyperbolic perturbations. It combines Gevrey regularity techniques with refined Siegel–Brjuno type estimates and a parameter-dependent resonance renormalization within a direct KAM framework to separate destruction and persistence phenomena for carefully chosen rotation numbers. The destruction result uses a precise lower bound on the Peierls barrier derived from a tightly controlled Gevrey bump, while persistence is established via a tree-expanded Lindstedt-series analysis with multi-scale resonance renormalization, under Brjuno-type arithmetical conditions. The paper also develops a nuanced arithmetic treatment of rotation numbers, demonstrating that destruction and persistence can be separated for different frequencies through bar{F}_{m,a} and bar{f}_{m,a} constructions, revealing a delicate interplay between perturbation structure, regularity, and number-theoretic properties.

Abstract

In his celebrated counterexample to the KAM theorem, Herman introduced a perturbation of an integrable system consisting of two components: a hyperbolic term and a bump function. He also remarked that it was unclear whether the bump function was truly necessary. In this note, we prove that the bump function is indeed necessary when more natural hyperbolic perturbations are considered. The proof of this necessity relies on an improved Siegel--Brjuno estimate and a parameter-dependent renormalization of resonances within the direct KAM method.

Herman's converse KAM mechanism revisited

TL;DR

This work revisits Herman's converse KAM mechanism in the setting of area-preserving twist maps, showing that a bump perturbation is essential when employing more natural hyperbolic perturbations. It combines Gevrey regularity techniques with refined Siegel–Brjuno type estimates and a parameter-dependent resonance renormalization within a direct KAM framework to separate destruction and persistence phenomena for carefully chosen rotation numbers. The destruction result uses a precise lower bound on the Peierls barrier derived from a tightly controlled Gevrey bump, while persistence is established via a tree-expanded Lindstedt-series analysis with multi-scale resonance renormalization, under Brjuno-type arithmetical conditions. The paper also develops a nuanced arithmetic treatment of rotation numbers, demonstrating that destruction and persistence can be separated for different frequencies through bar{F}_{m,a} and bar{f}_{m,a} constructions, revealing a delicate interplay between perturbation structure, regularity, and number-theoretic properties.

Abstract

In his celebrated counterexample to the KAM theorem, Herman introduced a perturbation of an integrable system consisting of two components: a hyperbolic term and a bump function. He also remarked that it was unclear whether the bump function was truly necessary. In this note, we prove that the bump function is indeed necessary when more natural hyperbolic perturbations are considered. The proof of this necessity relies on an improved Siegel--Brjuno estimate and a parameter-dependent renormalization of resonances within the direct KAM method.

Paper Structure

This paper contains 26 sections, 17 theorems, 187 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Herman's converse KAM mechanism
  3. On the necessity of the second perturbation
  4. Statement of main results
  5. On the destruction
  6. On the persistence
  7. On the arithmetics of the rotation number
  8. Remarks on Herman's counterexample
  9. Construction of the bump
  10. Gevrey-$\alpha$ function
  11. Construction of $\xi_m$
  12. Destruction of $\Gamma_\omega$ for $F_{m,a}$
  13. Basic tools from Aubry--Mather theory
  14. Minimal configuration
  15. Peierls's barrier
  16. ...and 11 more sections

Key Result

Theorem 1

Fix $a > 1$, $b > 0$ and $0 < \varepsilon \ll 1$. There exists $M_0 > 0$ such that for every $m \geq M_0$ and every rotation number $\omega$ satisfying where $n_m$ is defined in n_m, the map $F_{m,a}$ admits no invariant circle with rotation number $\omega$. Moreover, for any $\alpha > 1 + \frac{a}{\varepsilon}$ and $L \in (0, b)$,

Figures (3)

  • Figure 1: The hyperbolic perturbation in Herman's counterexample
  • Figure 2: Theoretical tools for handling $F_{N_m,a}$ and $f_{N_m,a}$
  • Figure 3: Schematic description of dynamics caused by two perturbations

Theorems & Definitions (35)

  • Remark 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 25 more