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.  built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed -manifold; the example in  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 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 and partially hyperbolic diffeomorphisms on 3-manifolds. Comment. Math. Helv. 92 (2017), no. 3, pp. 513–550DOI 10.4171/CMH/418