Commentarii Mathematici Helvetici


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Volume 92, Issue 3, 2017, pp. 513–550
DOI: 10.4171/CMH/418

Published online: 2017-07-27

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

Christian Bonatti[1] and Jinhua Zhang[2]

(1) Université de Bourgogne, Dijon, France
(2) Peking University, Beijing, China, and Université de Bourgogne, Dijon, France

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 $\mathrm {Diff}^{1}(\mathbb T^2)$ 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.

Keywords: Dehn twist, transverse foliations, partial hyperbolicity, transverse torus

Bonatti Christian, Zhang Jinhua: Transverse foliations on the torus $\mathbb T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds. Comment. Math. Helv. 92 (2017), 513-550. doi: 10.4171/CMH/418