Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms

We introduce the notion of $\mathit{\text{fibered}}\mathit{\text{lifted}}\mathit{\text{partially}}\mathit{\text{hyperbolic}}\mathit{\text{diffeomorphisms}}$ and we prove that any partially hyperbolic diifeomorphism isotopic to a fibered lifted one where the isotopy take place inside partially hyperbolic systems is $\mathit{\text{dynamically}}\mathit{\text{coherent.}}$ 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.