Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let M be a compact C-1 manifold which is invariant and normally hyperbolic with respect to a C-1 semiflow in a Banach space. Then in an epsilon-neighborhood of M there exist local C-1 center-stable and center-unstable manifolds W-cs (epsilon) and W-cu (epsilon), respectively. Here we show that W-cs (epsilon) and W-cu (epsilon) may each be decomposed into the disjoint union of C-1 submanifolds (leaves) in such a way that the semi ow takes leaves into leaves of the same collection. Furthermore, each leaf intersects M in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.