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Article Dans Une Revue Commentarii Mathematici Helvetici Année : 2017

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

Christian Bonatti
Jinhua Zhang
• Fonction : Auteur
• PersonId : 1017147

#### Résumé

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.

#### Domaines

Mathématiques [math]

### Dates et versions

hal-01591646 , version 1 (21-09-2017)

### Identifiants

• HAL Id : hal-01591646 , version 1
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• DOI :

### Citer

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

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