Transverse foliations on the torus T² and partially hyperbolic di eomorphisms on 3-manifolds

In this paper, we prove that given two C¹ foliations F and G on T² which are transverse, there exists a non-null homotopic loop {ℙ_t}_t∈ in Diff¹(T²) such that ℙ_t (F) ? for every t ∈ [0, 1] and ℙ₀ ℙ₁ = Id. As a direct consequence, we get a general process for building new partially hyperbolic di eomorphisms on closed 3-manifolds. Bonatti et al. [4] built a new example of dynamically coherent non-transitive partially hyperbolic di eomorphism 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 nontransitive 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 di eomorphisms.