{
  "id": "0801.1639",
  "version": "v2",
  "published": "2008-01-10T17:37:53.000Z",
  "updated": "2009-09-30T18:44:20.000Z",
  "title": "Itineraries of rigid rotations and diffeomorphisms of the circle",
  "authors": [
    "David Richeson",
    "Paul Winkler",
    "Jim Wiseman"
  ],
  "comment": "Added error estimates in response to referees' comments",
  "categories": [
    "math.DS"
  ],
  "abstract": "We examine the itinerary of $0\\in S^{1}=\\R/\\Z$ under the rotation by $\\alpha\\in\\R\\bs\\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\\subset S^{1}$, a finite union of closed intervals, can we recover $\\alpha$ and $I$? We prove that the itineraries do determine $\\alpha$ and $I$ up to certain equivalences. Then we present elementary methods for finding $\\alpha$ and $I$. Moreover, if $g:S^{1}\\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-09-30T18:44:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "37E45",
      "37B10"
    ],
    "keywords": [
      "rigid rotations",
      "irrational rotation number",
      "finite union",
      "orbit itinerary",
      "orientation preserving diffeomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}