Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Abstract

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