{
  "id": "1710.02371",
  "version": "v1",
  "published": "2017-10-06T12:45:58.000Z",
  "updated": "2017-10-06T12:45:58.000Z",
  "title": "Dynamics of the scenery flow and conical density theorems",
  "authors": [
    "Antti Käenmäki"
  ],
  "categories": [
    "math.DS",
    "math.HO"
  ],
  "abstract": "Conical density theorems are used in the geometric measure theory to derive geometric information from given metric information. The idea is to examine how a measure is distributed in small balls. Finding conditions that guarantee the measure to be effectively spread out in different directions is a classical question going back to Besicovitch (1938) and Marstrand (1954). Classically, conical density theorems deal with the distribution of the Hausdorff measure. The process of taking blow-ups of a measure around a point induces a natural dynamical system called the scenery flow. Relying on this dynamics makes it possible to apply ergodic-theoretical methods to understand the statistical behavior of tangent measures. This approach was initiated by Furstenberg (1970, 2008) and greatly developed by Hochman (2010). The scenery flow is a well-suited tool to address problems concerning conical densities. In this survey, we demonstrate how to develop the ergodic-theoretical machinery around the scenery flow and use it to study conical density theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-06T12:45:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37A10",
      "28A75",
      "28A33"
    ],
    "keywords": [
      "scenery flow",
      "study conical density theorems",
      "address problems concerning conical densities",
      "geometric measure theory",
      "conical density theorems deal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}