{
  "id": "1911.11996",
  "version": "v1",
  "published": "2019-11-27T07:32:48.000Z",
  "updated": "2019-11-27T07:32:48.000Z",
  "title": "Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits",
  "authors": [
    "Matthew D. Kvalheim",
    "Shai Revzen"
  ],
  "comment": "26 pages, 1 figure, comments welcome",
  "categories": [
    "math.DS",
    "math.OC"
  ],
  "abstract": "We consider $C^1$ dynamical systems having a globally attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for $C^{k,\\alpha}_{\\text{loc}}$ globally defined linearizing semiconjugacies, of which Koopman eigenfunctions are a special case. Our main results both generalize and sharpen Sternberg's $C^k$ linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Additional corollaries include existence and uniqueness results for $C^{k,\\alpha}_{\\text{loc}}$ Koopman eigenfunctions, including a complete classification of $C^\\infty$ eigenfunctions assuming a $C^\\infty$ dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of \"principal Koopman eigenfunctions\" which generalizes the definition of Mohr and Mezi\\'{c} for linear systems, and which includes the notions of \"isostables\" and \"isostable coordinates\" appearing in work by Ermentrout, Mauroy, Mezi\\'{c}, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) \"pullback algebra\" defined in \\cite{mohr2016koopman} is unique under certain conditions. We also discuss the limit used to define the \"faster\" isostable coordinates in \\cite{wilson2018greater,monga2019phase} in light of our main results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-27T07:32:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C05",
      "37C10",
      "37C15",
      "37C27",
      "34C15",
      "34C20",
      "37D99"
    ],
    "keywords": [
      "global koopman eigenfunctions",
      "stable fixed points",
      "periodic orbit",
      "attracting hyperbolic fixed point",
      "isostable coordinates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}