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

Abstract

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 ${\Phi }_t(\mathcal{F})⋔\mathcal{G}$ 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.