{
  "id": "1710.07555",
  "version": "v1",
  "published": "2017-10-20T14:47:21.000Z",
  "updated": "2017-10-20T14:47:21.000Z",
  "title": "Some observations on Käenmäki measures",
  "authors": [
    "Ian D. Morris"
  ],
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "In this note we investigate some properties of equilibrium states of affine iterated function systems, sometimes known as K\\\"aenm\\\"aki measures. We give a simple sufficient condition for K\\\"aenm\\\"aki measures to have a gap between certain specific pairs of Lyapunov exponents, partially answering a question of B. B\\'ar\\'any, A. K\\\"aenm\\\"aki and H. Koivusalo. We also give sharp bounds for the number of ergodic K\\\"aenm\\\"aki measures in dimensions up to 4, answering a question of J. Bochi and the author within this range of dimensions. Finally, we pose an open problem on the Hausdorff dimension of self-affine measures which may be reduced to a statement concerning semigroups of matrices in which a particular weighted product of absolute eigenvalues is constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-20T14:47:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37D35",
      "37H15"
    ],
    "keywords": [
      "käenmäki measures",
      "observations",
      "affine iterated function systems",
      "simple sufficient condition",
      "equilibrium states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}