A partially hyperbolic diffeomorphism $f:M\to M$ with splitting $TM=E^{ss}\oplus E^c\oplus E^{uu}$ is said to be dynamically coherent if the center bundle $E^c$ integrates to an invariant foliation of $M.$
Let $f :M\to M$ be a dynamically coherent partially hyperbolic diffeomorphism, is it still dynamically coherent when perturbed? Classical results by Hirch-Pugh-Shub give sufficient conditions for this question to have an affirmative answer. But what if instead of perturbing one takes a path of partially hyperbolic diffeomorphisms (without modifying the dimensions of the fiber bundles), is it still dynamically coherent all along the way? Recently S. Martinchich proved that it is so when the center leaves "are fixed" (and one-dimensional) by the diffeomorphism. We will give an affirmative answer also in an opposite case: when the dynamics of the center leaves is "hyperbolic", no matter the dimension of the center bundle. This is part of Luis Piñeyrúa's PhD thesis. | 
