{
  "id": "2502.08315",
  "version": "v1",
  "published": "2025-02-12T11:31:51.000Z",
  "updated": "2025-02-12T11:31:51.000Z",
  "title": "Shadowing for Infinite Dimensional Dynamical Systems",
  "authors": [
    "José M. Arrieta",
    "Alexandre N. Carvalho",
    "Carlos R. Takaessu Jr"
  ],
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "In this paper we extend some results about Shadowing Lemma there are known on finite dimensional compact manifolds without border and $\\mathbb{R}^n$, to an infinite dimensional space. In fact, we prove that if $\\{\\mathcal{T}(t):t\\ge 0\\}$ is a Morse-Smale semigroup defined in a Hilbert space with global attractor $\\mathcal{A}$, then $\\mathcal{T}(1)|_{\\mathcal{A}}:\\mathcal{A}\\to \\mathcal{A} $ admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $\\mathcal{U}\\supset\\mathcal{A}$ of the global attractor, the map $\\mathcal{T}(1)|_{\\mathcal{U}}:\\mathcal{U}\\to \\mathcal{U}$ has the H\\\"{o}lder-Shadowing property. As applications, we obtain new results related to the structural stability of Morse-Smale semigroups defined in Hilbert spaces and continuity of global attractors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-12T11:31:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite dimensional dynamical systems",
      "global attractor",
      "hilbert space",
      "morse-smale semigroup",
      "finite dimensional compact manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}