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

Abstract : In this paper, we prove that given two $C^1$ foliations $F$ and $G$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop ${\{\Phi_t\}_{t\in[0,1]}}$ in $\mathrm {Diff}^{1}(\mathbb T^2)$ such that ${\Phi_t(\mathcal{F})\pitchfork \mathcal{G}}$ for every $t\in[0,1]$, and $\Phi_0=\Phi_1= \mathrm {Id}$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed $3$-manifolds. [4] built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold; the example in [4] 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.
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Commentarii Mathematici Helvetici, European Mathematical Society, 2017, 92 (3), pp.513 - 550. 〈10.4171/CMH/418〉
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01591646
Contributeur : Imb - Université de Bourgogne <>
Soumis le : jeudi 21 septembre 2017 - 16:49:32
Dernière modification le : jeudi 11 janvier 2018 - 06:12:20

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Christian Bonatti, Jinhua Zhang. Transverse foliations on the torus $\mathbb T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds. Commentarii Mathematici Helvetici, European Mathematical Society, 2017, 92 (3), pp.513 - 550. 〈10.4171/CMH/418〉. 〈hal-01591646〉

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