Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms

We introduce the notion of fibered lifted partially hyperbolic diffeomorphisms and we prove that any partially hyperbolic diffeomorphism isotopic to a fibered lifted one where the isotopy take place inside partially hyperbolic systems is dynamically coherent. Moreover we prove some global stability result: every two partially hyperbolic diffeomorphisms in the same connected component of a fibered lifted partially hyperbolic diffeomorphisms are leaf conjugate.