# Dynamical foliations for system isotopic to Anosov diffeomophisms

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We discuss the existence of foliations that are invariant under the dynamics for systems that are isotopic to Anosov diffeomorphisms. Specifically, we examine partially hyperbolic diffeomorphisms with one dimensional center that are isotopic to a hyperbolic toral automorphism and contained in a connected component. We show in this case there is a center foliation. We will also discuss more general cases where there is a weak form of hyperbolicity called a dominated splitting. This is joint work with Jerome Buzzi, Rafael Potrie, and Martin Sambarino.