{
  "id": "2403.19958",
  "version": "v1",
  "published": "2024-03-29T03:27:30.000Z",
  "updated": "2024-03-29T03:27:30.000Z",
  "title": "Uniformity of geodesic flow in non-integrable 3-manifolds",
  "authors": [
    "J. Beck",
    "W. W. L. Chen",
    "Y. Yang"
  ],
  "comment": "44 pages, 32 figures",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Almost nothing is known concerning the extension of $3$-dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus $[0,1)^3$ to non-integrable finite polycube translation $3$-manifolds. In the special case when a finite polycube translation $3$-manifold is the cartesian product of a finite polysquare translation surface with the unit torus $[0,1)$, we have developed a splitting method with which we can make some progress. This is a somewhat restricted system, in the sense that one of the directions is integrable. We then combine this with a split-covering argument to extend our results to some other finite polycube translation $3$-manifolds which satisfy a rather special condition and where none of the $3$ directions is integrable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-29T03:27:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E35",
      "11K38"
    ],
    "keywords": [
      "geodesic flow",
      "finite polysquare translation surface",
      "unit torus",
      "uniformity",
      "dimensional kronecker-weyl equidistribution theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}