00-236 Mohamed S. ElBialy

Sub-Stable and Weak-Stable Manifolds Associated with Finitely Non-Resonant Spectral Subspaces For Maps of a Banach Space (192K, LaTeX 2e) May 19, 00

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Abstract. In this work we study $C^{k,\gd}, 2 \leq k \leq \infty, 0\leq \gd \leq 1,$ maps of a Banach space near a fixed point. We show the existence and uniqueness of a class of $C^{k,\gd}$ local invariant sub-manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non-resonance condition of order $L\leq k$ and an overriding condition of order $L\leq k$ (condition (3) of Theorem \ref{TH:1}). We study the dependence of these invariant manifolds on a parameter that lies in a Banach space. We also show that a $C^{k,\gd}$ local weak-stable manifold that satisfies these two conditions is unique in the class of $C^{k,\gd}$ maps. The uniqueness is due to the fact that our method does not require a cut-off function. An infinite dimensional Banach space does not always admit smooth cut-off functions.

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