Persistence of Overflowing Manifolds for Semiflow

This paper, which is a sequel to a previous one [4] by the same authors, is devoted to the persistence of overflowing manifolds and inflowing manifolds for a semiflow in a Banach space. We consider a C1 semiflow defined on a Banach space X; that is, it is continuous on [0,∞)×X , and for each t ≥ 0, T t : X → X is C1, and T t ◦ T (x) = T t+s(x) for all t, s ≥ 0 and x ∈ X . A typical example is the solution operator for a differential equation. In [4] we proved that a compact, normally hyperbolic, invariant manifold M persists under small C1 perturbations in the semiflow. We also showed that in a neighborhood of M , there exist a center-stable manifold and a center-unstable manifold that intersect in the manifold M . In [4] the compactness and invariance of the manifold M were important assumptions. In the present paper, we study the more general case where the manifold M is overflowing (“negatively invariant and the semiflow crosses the boundary transversally”) or inflowing (“positively invariant and the semiflow crosses the boundary transversally”). We do not assume that M is compact or finite-dimensional. Also, M is not necessarily an imbedded manifold, but an immersed manifold. As an example, a local unstable manifold of an equilibrium point is an overflowing manifold. In brief, our main results on the overflowing manifolds may be summarized as follows (the precise statements are given in Section 2). We assume that the immersed manifoldM does not twist very much locally,M is covered by the image under T t of a subset a positive distance away from boundary, DT t contracts along the normal direction and does so more strongly than it does along the tangential direction, and DT t has a certain uniform continuity in a neighborhood of M . If the C1 perturbation T t of T t is sufficiently close to T t, then T t has a unique C1 immersed overflowing manifold M nearM . Furthermore, if T t isCk and a spectral gap condition holds, then M isCk. Similar results for inflowing manifolds are also obtained and given in Section 7.

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