00-237 Mohamed S. ElBialy

Local Contractions of Banach Spaces and Spectral Gap Conditions (111K, LaTeX 2e) May 19, 00

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Abstract. In this work we study the linearization problem for a $C^{k,1}, k\geq 1,$ contraction of a Banach space $E$ near a fixed point which satisfies a spectral gap condition and a narrow band condition both of order $k$. We also assume that the part of the spectrum in each band satisfies a finite nonresonant condition of order $k$ relative to itself together with the part that lies in the larger bands. We show that there is a $C^{k,\gb}$ linearization for sufficiently small $\gb > 0$. We give a precise estimate on $\gb$ in terms of the gap and band conditions.

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