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 semiflow 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.