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<p>We show the following dichotomy for a linear parabolic <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">${\mathbb{Z}}^{2}$</EquationSource> </InlineEquation>-action <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>L</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\rho _{L}$</EquationSource> </InlineEquation> on the torus with at least one step-2 generator: (i) Any affine <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">${\mathbb{Z}}^{2}$</EquationSource> </InlineEquation>-action with linear part <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>L</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\rho _{L}$</EquationSource> </InlineEquation> has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">${\mathbb{Z}}^{2}$</EquationSource> </InlineEquation>-action with linear part <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>L</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\rho _{L}$</EquationSource> </InlineEquation> is KAM-rigid under volume preserving perturbations.</p>

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KAM-rigidity for parabolic affine Abelian actions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月23日

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  • Danijela Damjanović,
  • Bassam Fayad,
  • Maria Saprykina

摘要

We show the following dichotomy for a linear parabolic Z 2 ${\mathbb{Z}}^{2}$ -action ρ L $\rho _{L}$ on the torus with at least one step-2 generator: (i) Any affine Z 2 ${\mathbb{Z}}^{2}$ -action with linear part ρ L $\rho _{L}$ has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine Z 2 ${\mathbb{Z}}^{2}$ -action with linear part ρ L $\rho _{L}$ is KAM-rigid under volume preserving perturbations.