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 $\diff^{ 1}(\T^2)$ such that $\Phi_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, 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. \cite{ BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold, the example in \cite{ BPP} 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.
Categories: math.DS