{
  "id": "2208.10115",
  "version": "v1",
  "published": "2022-08-22T07:40:58.000Z",
  "updated": "2022-08-22T07:40:58.000Z",
  "title": "Invariant tori for area-preserving maps with ultra-differentiable perturbation and Liouvillean frequency",
  "authors": [
    "Hongyu Cheng",
    "Shimin Wang",
    "Fenfen Wang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove the existence of invariant tori to the area-preserving maps defined on $ \\mathbb{R}^2\\times\\mathbb{T} $ \\begin{equation*} \\bar{x}=F(x,\\theta), \\qquad \\bar{\\theta}=\\theta+\\alpha\\, \\,(\\alpha\\in \\mathbb{R}\\setminus\\mathbb{Q}), \\end{equation*} where $ F $ is closed to a linear rotation, and the perturbation is ultra-differentiable in $ \\theta\\in \\mathbb{T},$ which is very closed to $C^{\\infty}$ regularity. Moreover, we assume that the frequency $\\alpha$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on $\\alpha$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-22T07:40:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "area-preserving maps",
      "invariant tori",
      "liouvillean frequency",
      "ultra-differentiable perturbation",
      "linear rotation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}