Local Contractions of Banach Spaces and Spectral Gap Conditions

We study the linearization problem for a Ck, 1, k⩾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 non-resonant condition of order k relative to itself together with the part that lies in the larger bands. We show that there is a Ck, β linearization for sufficiently small β>0. We give a precise estimate on β in terms of the gap and band conditions.