{
  "id": "1110.4354",
  "version": "v3",
  "published": "2011-10-19T19:07:58.000Z",
  "updated": "2012-05-14T21:19:02.000Z",
  "title": "Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications",
  "authors": [
    "Micka ël D. Chekroun",
    "Nathan E. Glatt-Holtz"
  ],
  "comment": "To appear in Communications in Mathematical Physics",
  "categories": [
    "math.DS",
    "math-ph",
    "math.AP",
    "math.FA",
    "math.MP"
  ],
  "abstract": "In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\\{S(t)\\}_{t \\geq 0}$. Suppose that $\\S(t)\\}_{t \\geq 0}$ possesses a global attractor $\\mathcal{A}$. We show that, for any generalized Banach limit $\\underset{T \\rightarrow \\infty}{\\rm{LIM}}$ and any distribution of initial conditions $\\mathfrak{m}_0$, that there exists an invariant probability measure $\\mathfrak{m}$, whose support is contained in $\\mathcal{A}$, such that $$ \\int_{X} \\phi(x) d\\mathfrak{m} (x) = \\underset{T\\to \\infty}{\\rm{LIM}} \\frac{1}{T}\\int_0^T \\int_X \\phi(S(t) x) d \\mathfrak{m}_0(x) d t, $$ for all observables $\\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\\{S(t)\\}_{t \\geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-14T21:19:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K40",
      "35B41",
      "35B40",
      "35Q35",
      "37L40",
      "37L50",
      "37L05",
      "37N10",
      "47H20",
      "60B05",
      "76A10"
    ],
    "keywords": [
      "invariant measures",
      "dissipative dynamical systems",
      "abstract results",
      "generalized banach limit",
      "applications"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-012-1515-y",
      "journal": "Communications in Mathematical Physics",
      "year": 2012,
      "month": "Dec",
      "volume": 316,
      "number": 3,
      "pages": 723
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012CMaPh.316..723C"
    }
  }
}