{
  "id": "1602.04355",
  "version": "v1",
  "published": "2016-02-13T16:56:24.000Z",
  "updated": "2016-02-13T16:56:24.000Z",
  "title": "Transverse foliations on the torus $\\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds",
  "authors": [
    "Christian Bonatti",
    "Jinhua Zhang"
  ],
  "comment": "34 pages, 7 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we prove that given two $C^1$ foliations $\\mathcal{F}$ and $\\mathcal{G}$ on $\\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\\{\\Phi_t\\}_{t\\in[0,1]}$ in $\\diff^{1}(\\T^2)$ such that $\\Phi_t(\\calF)\\pitchfork \\calG$ for every $t\\in[0,1]$, and $\\Phi_0=\\Phi_1= Id$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed $3$-manifolds. \\cite{BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold, the example in \\cite{BPP} is obtained by composing the time $t$ map, $t>0$ large enough, of a very specific non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result shows that the same construction holds starting with any non-transitive Anosov flow on an oriented $3$-manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-13T16:56:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "transverse foliations",
      "dehn twist",
      "transverse torus",
      "specific non-transitive anosov flow",
      "non-null homotopic loop"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160204355B"
    }
  }
}