  We consider hyperbolic and partially hyperbolic diffeomorphisms on compact
manifolds. Associated with invariant foliation of these systems, we define some
topological invariants and show certain relationships between these topological
invariants and the geometric and Lyapunov growths of these foliations. As an
application, we show examples of systems with persistent non- absolute
continuous center and weak unstable foliations. This generalizes the remarkable
results of Shub and Wilkinson to cases where the center manifolds are not
compact.
