{
  "id": "0710.2836",
  "version": "v1",
  "published": "2007-10-15T18:14:35.000Z",
  "updated": "2007-10-15T18:14:35.000Z",
  "title": "Topological entropies of equivalent smooth flows",
  "authors": [
    "Wenxiang Sun",
    "Todd Young",
    "Yunhua Zhou"
  ],
  "journal": "Trans. A.M.S. 361 (2009), 3071-3082",
  "categories": [
    "math.DS"
  ],
  "abstract": "Two flows defined on a smooth manifold are equivalent if there exists a homeomorphism of the manifold that sends each orbit of one flow onto an orbit of the other flow while preserving the time orientation. The topological entropy of a flow is defined as the entropy of its time-1 map. While topological entropy is an invariant for equivalent homeomorphisms, finite non-zero topological entropy for a flow cannot be an invariant because its value is affected by time reparameterization. However, 0 and $\\infty$ topological entropy are invariants for equivalent flows without fixed points. In equivalent flows with fixed points there exists a counterexample, constructed by Ohno, showing that neither 0 nor $\\infty$ topological entropy is preserved by equivalence. The two flows constructed by Ohno are suspensions of a transitive subshift and thus are not differentiable. Note that a differentiable flow on a compact manifold cannot have $\\infty$ entropy. These facts led Ohno in 1980 to ask the following: \"Is 0 topological entropy an invariant for equivalent differentiable flows?\" In this paper, we construct two equivalent $C^\\infty$ smooth flows with a singularity, one of which has positive topological entropy while the other has zero topological entropy. This gives a negative answer to Ohno's question in the class $C^\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-15T18:14:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C15",
      "34C28",
      "37A10"
    ],
    "keywords": [
      "equivalent smooth flows",
      "equivalent flows",
      "finite non-zero topological entropy",
      "fixed points",
      "differentiable flow"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.2836S"
    }
  }
}