Transverse foliations on the torus T-2 and partially hyperbolic diffeomorphisms on 3-manifolds

In this paper, we prove that given two C-1 foliations F and G on T-2 which are transverse, there exists a non-null homotopic loop {Phi(t)}t is an element of [0, 1] in Diff(1)(T-2) such that Phi(t) (F) G for every t is an element of [0, 1], and Phi(0) = Phi(1) = Id. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed 3-manifolds. Bonatti et al. [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.