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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{1-\varepsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>ε</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-topology. This result is sharp, as evidenced by the persistence of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-invariant graphs under <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-perturbations guaranteed by the normally hyperbolic invariant manifold theorem.</p>

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On the destruction of invariant lagrangian graphs for conformal symplectic twist maps

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Alfonso Sorrentino,
  • Lin Wang

摘要

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 \(C^0\) C 0 -invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the \(C^{1-\varepsilon }\) C 1 - ε -topology. This result is sharp, as evidenced by the persistence of \(C^1\) C 1 -invariant graphs under \(C^1\) C 1 -perturbations guaranteed by the normally hyperbolic invariant manifold theorem.