{
  "id": "1510.01605",
  "version": "v1",
  "published": "2015-10-06T14:52:03.000Z",
  "updated": "2015-10-06T14:52:03.000Z",
  "title": "Mean dimension of $\\mathbb{Z}^k$-actions",
  "authors": [
    "Yonatan Gutman",
    "Elon Lindenstrauss",
    "Masaki Tsukamoto"
  ],
  "comment": "44 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $\\mathbb{Z}^k$-action on a compact metric space $X$, we study the following three problems closely related to mean dimension. (1) When is $X$ isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy $h_{\\mathrm{top}}(X)$ is infinite. How much topological entropy can be detected if one considers $X$ only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed $X$ into the $\\mathbb{Z}^k$-shift on the infinite dimensional cube $([0,1]^D)^{\\mathbb{Z}^k}$? These were investigated for $\\mathbb{Z}$-actions in [Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes \\'Etudes Sci. Publ. Math. \\textbf{89} (1999) 227-262], but the generalization to $\\mathbb{Z}^k$ remained an open problem. When $X$ has the marker property, in particular when $X$ has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3). A key ingredient is a new method to continuously partition every orbit into good pieces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-06T14:52:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B40",
      "54F45"
    ],
    "keywords": [
      "mean dimension",
      "finite entropy systems",
      "aperiodic minimal factor",
      "infinite dimensional cube",
      "compact metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151001605G"
    }
  }
}