Invariant Tori for Area-Preserving Maps with Ultra-differentiable Perturbation and Liouvillean Frequency

• Hongyu Cheng ^[3] ; Fenfen Wang ^[1] ; Shimin Wang ^[2]
  1. [1] Sichuan Normal University

     Sichuan Normal University

     China

     
  2. [2] Shandong University

     Shandong University

     China

     
  3. [3] Tiangong University

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01143-4
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• Resumen

  • We prove the existence of invariant tori to the area-preserving maps defined on R2 ×T x = F(x, θ ), θ = θ + α (α ∈ R\Q), where F is related to a linear rotation, and the perturbation is ultra-differentiable in θ ∈ T, which is very closed to C∞ regularity. Moreover, we assume that the frequency α is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on α. Thus, both the difficulties from the ultra-differentiability of the perturbation and Liouvillean frequency will appear in this work. The proof of the main result is based on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preserving maps with some new techniques.

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