JournalscmhVol. 92, No. 3pp. 513–550

Transverse foliations on the torus T2\mathbb T^2 and partially hyperbolic diffeomorphisms on 3-manifolds

  • Christian Bonatti

    Université de Bourgogne, Dijon, France
  • Jinhua Zhang

    Peking University, Beijing, China, and Université de Bourgogne, Dijon, France
Transverse foliations on the torus $\mathbb T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds cover

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Abstract

In this paper, we prove that given two C1C^1 foliations F\mathcal{F} and G\mathcal{G} on T2\mathbb{T}^2 which are transverse, there exists a non-null homotopic loop {Φt}t[0,1]{\{\Phi_t\}_{t\in[0,1]}} in Diff1(T2)\mathrm {Diff}^{1}(\mathbb T^2) such that Φt(F)G{\Phi_t(\mathcal{F})\pitchfork \mathcal{G}} for every t[0,1]t\in[0,1], and Φ0=Φ1=Id\Phi_0=\Phi_1= \mathrm {Id}.

As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed 33-manifolds. [4] built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed 33-manifold; the example in [4] is obtained by composing the time tt map, t>0t>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.

Cite this article

Christian Bonatti, Jinhua Zhang, Transverse foliations on the torus T2\mathbb T^2 and partially hyperbolic diffeomorphisms on 3-manifolds. Comment. Math. Helv. 92 (2017), no. 3, pp. 513–550

DOI 10.4171/CMH/418