Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Bates, Peter; Lu, Kening; Zeng, Chongchun

doi:10.1090/S0002-9947-00-02503-4

Invariant foliations near normally hyperbolic invariant manifolds for semiflows
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by Peter W. Bates, Kening Lu and Chongchun Zeng

Trans. Amer. Math. Soc. 352 (2000), 4641-4676

DOI: https://doi.org/10.1090/S0002-9947-00-02503-4

Published electronically: June 14, 2000

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Abstract:

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.

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