{
  "id": "1803.02440",
  "version": "v1",
  "published": "2018-03-06T22:15:10.000Z",
  "updated": "2018-03-06T22:15:10.000Z",
  "title": "A shift map with a discontinuous entropy function",
  "authors": [
    "Christian Wolf"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f:X\\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $\\mu\\mapsto h_\\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $\\phi:X\\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\\Phi:X\\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\\Phi$ with a discontinuous localized entropy function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-06T22:15:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "37B10",
      "37C40"
    ],
    "keywords": [
      "discontinuous entropy function",
      "shift map",
      "dimensional lipschitz continuous potential",
      "compact metric space",
      "discontinuous localized entropy function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}