Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms (Q6564151)

In the paper, the authors introduce the notion of \textit{fibered lifted partially hyperbolic diffeomorphisms}. More precisely, a dynamically coherent partially hyperbolic diffeomorphism \({f: M\to M}\) is said to be \textit{fibered lifted} if it satisfies the following conditions:\N\N(1) The foliations \(\tilde{\mathcal{W}}^{cs}_f\) and \(\tilde{\mathcal{W}}^{u}_f\) (as well as \(\tilde{\mathcal{W}}^{cu}_f\) and \(\tilde{\mathcal{W}}^{s}_f\)) have global product structure in the universal cover \(\tilde{M}\);\N\N(2) The quotient space \(\tilde{M}/ \tilde{\mathcal{W}}^c_f\) is a topological manifold and the induced map \[{\tilde{f}_c : \tilde{M}/ \tilde{\mathcal{W}}^c_f\to \tilde{M}/\tilde{\mathcal{W}}^c_f}\] is a hyperbolic homeomorphism.\N\NThe main result states that any partially hyperbolic diffeomorphism isotopic to a fibered lifted one is dynamically coherent. The isotopy takes place inside partially hyperbolic systems. The authors also discuss some related properties and consider some examples.