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On the Destruction of Invariant Lagrangian Graphs for Conformal Symplectic Twist Maps

Alfonso Sorrentino, Lin Wang

TL;DR

The paper establishes that invariant $C^0$-Lagrangian graphs for integrable dissipative (conformal) twist maps are highly fragile: arbitrarily small perturbations in the $C^{1-\varepsilon}$ topology can destroy all such graphs. The authors extend Herman’s a-posteriori framework to the dissipative, conformal-symplectic setting and, using Jackson’s approximation, construct perturbations with precisely controlled regularity and degree to violate the invariant-graph condition. The results are sharp due to normally hyperbolic invariant manifold theory, which guarantees persistence of $C^1$-invariant graphs under $C^1$-perturbations, thus delineating the boundary between stability and destruction. The higher-dimensional generalization to $\mathbb{T}^d\times\mathbb{R}^d$ mirrors the 1D approach, yielding a d-dimensional analogue of Herman’s formula and a corresponding perturbation construction, with implications for the structure of Birkhoff attractors in dissipative dynamics.

Abstract

In this article we investigate the fragility of invariant Lagrangian graphs for dissipative maps, focusing on their destruction under small perturbations. Inspired by Herman's work on conservative systems, we prove that all $C^0$-invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the $C^{1-\varepsilon}$-topology. This result is sharp, as evidenced by the persistence of $C^1$-invariant graphs under $C^1$-perturbations guaranteed by the normally hyperbolic invariant manifold theorem.

On the Destruction of Invariant Lagrangian Graphs for Conformal Symplectic Twist Maps

TL;DR

The paper establishes that invariant -Lagrangian graphs for integrable dissipative (conformal) twist maps are highly fragile: arbitrarily small perturbations in the topology can destroy all such graphs. The authors extend Herman’s a-posteriori framework to the dissipative, conformal-symplectic setting and, using Jackson’s approximation, construct perturbations with precisely controlled regularity and degree to violate the invariant-graph condition. The results are sharp due to normally hyperbolic invariant manifold theory, which guarantees persistence of -invariant graphs under -perturbations, thus delineating the boundary between stability and destruction. The higher-dimensional generalization to mirrors the 1D approach, yielding a d-dimensional analogue of Herman’s formula and a corresponding perturbation construction, with implications for the structure of Birkhoff attractors in dissipative dynamics.

Abstract

In this article we investigate the fragility of invariant Lagrangian graphs for dissipative maps, focusing on their destruction under small perturbations. Inspired by Herman's work on conservative systems, we prove that all -invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the -topology. This result is sharp, as evidenced by the persistence of -invariant graphs under -perturbations guaranteed by the normally hyperbolic invariant manifold theorem.

Paper Structure

This paper contains 12 sections, 8 theorems, 100 equations.

Table of Contents

  1. Introduction and main results
  2. $1$-dimensional case
  3. $d$-dimensional case
  4. Proof of Theorem \ref{['Mt']}
  5. Herman's formula for dissipative twist maps
  6. Case 1
  7. Case 2
  8. Construction of the perturbation
  9. Proof of Theorem \ref{['Mt2']}
  10. Herman's formula for conformal symplectic maps on $\mathbb{T}^d$
  11. Construction of the perturbation
  12. On multivariate Jackson's approximation theorem

Key Result

Theorem 1

Given $\lambda\in (0,1)$ and $0<\varepsilon\ll 1$, there exists a sequence of trigonometric polynomials $\{\varphi^{\lambda}_n\}_{n\in \mathbb{N}}$ such that all of the $C^0$-invariant graphs for the family of $\{F_{\lambda,\alpha}\}_{\alpha\in\mathbb{R}^2}$ can be destroyed by perturbing the maps w

Theorems & Definitions (22)

  • Definition 1.1: Dissipative twist map of the cylinder
  • Remark 1.2
  • Definition 1.3: $C^0$-invariant graph
  • Remark 1.4
  • Theorem 1
  • Definition 1.5: conformal symplectic twist map
  • Remark 1.6
  • Definition 1.7: $C^0$-invariant Lagrangian graph
  • Remark 1.8
  • Remark 1.9
  • ...and 12 more