Transverse foliations on the torus $\T^2$ 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 $\diff^{1}(\T^2)$ such that $Φ_t(\calF)\pitchfork \calG$ 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.

34 pages, 7 figures

Cited by in corpus (1)

• Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds