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Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds

Christian Bonatti, Jinhua Zhang

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 $\{Φ_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that $Φ_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $Φ_0=Φ_1= Id$. 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.

Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds

Abstract

In this paper, we prove that given two foliations and on which are transverse, there exists a non-null homotopic loop in such that for every , and . As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed -manifolds. \cite{BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed -manifold, the example in \cite{BPP} is obtained by composing the time map, 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 -manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms.

Paper Structure

This paper contains 27 sections, 39 theorems, 29 equations, 7 figures.

Table of Contents

  1. Introduction and statement of the main results
  2. Pair of transverse foliations on ${\mathbb T}^2$
  3. Dehn twists and pairs of transverse $2$-foliations on $3$-manifolds
  4. Building partially hyperbolic diffeomorphisms on $3$-manifolds
  5. Definition of partially hyperbolic diffeomorphisms
  6. Definition of Anosov flows
  7. Transverse tori
  8. Statement of our main result
  9. Preliminary: foliations on the torus and its classification
  10. Complete transversal
  11. Reeb components
  12. Translation and rotation numbers
  13. Foliations without compact leaves on the annulus
  14. Existence of a complete transversal curve for two transverse foliations without parallel compact leaves
  15. Deformation of a foliation along its transverse foliation
  16. ...and 12 more sections

Key Result

Theorem A

Let $\mathcal{F}$ and $\mathcal{G}$ be two $C^1$ one-dimensional foliations on $\mathbb{T}^2$ and they are transverse. Then there exists a continuous family $\{\Phi_t\}_{t\in[0,1]}$ of $C^1$ diffeomorphisms on $\mathbb{T}^2$ such that

Figures (7)

  • Figure 1:
  • Figure 2: Reeb component
  • Figure 3: In the first figure: the dash line is the transversal $\gamma$; the dash and real arrows on the circle pointing outside give the directions of $\mathcal{G}$ and $\mathcal{F}$ respectively. The second and the third figure show the good choice of curve and bad choice of curve respectively.
  • Figure 4: The dash line is the transversal $\gamma$. The dash and real arrows on the circle pointing outside give the directions of $\mathcal{G}$ and $\mathcal{F}$ respectively.
  • Figure 5: The light line, the dark line and dash line denote the leaves of $\tilde{\mathcal{G}}$, $\tilde{\mathcal{E}}$ and $\tilde{\mathcal{F}}$ respectively.
  • ...and 2 more figures

Theorems & Definitions (84)

  • Theorem A
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
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  • Remark \oldthetheorem
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  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 74 more