A RESULT ON CLASS-C 1 LINEARIZATION OF CONTRACTIONS IN INFINITE DIMENSIONS. HILDEBRANDO M. RODRIGUES † AND J. SOLA-MORALES ‡ 1. Motivation. This note is a short presentation of our results in [5]. We start explaining, as a motivating example, a situation where a result of C 1 -linearization in infinite dimensions was needed and used. In the paper [2] it was proved that for some nonlinearities f (x, u) and some small values of α> 0 the global attractor of the dynamical system defined in H 1 (0,π) × L 2 (0,π) by the second order initial-boundary value problem u tt +2αu t = u xx + f (x, u), 0 <x<π u x (0) = u x (π)=0 is not contained on any finite-dimensional invariant manifold of class C 1 . In one of the steps of the proof it was needed to prove that for the case f (x, u) ≡ f (u) with f (0) = 0 and f ′ (0) < 0 there are only countable many finite-dimensional invariant manifolds of class C 1 containing the (assimptotically stable) equilibrium point (u, u t ) = (0, 0). This fact was proved by linearization, that is by showing that under an abstract change of variables of class C 1 in a neighborhood of (0, 0) ∈ H 1 (0,π) × L 2 (0,π), the equation turned into its linear part v tt +2αv t = v xx + f ′ (0)v, 0 <x<π v x (0) = v x (π)=0. †Partially supported by Fapesp, CNPq and CAPES/MECD 023/01, Brazil. ‡Partially supported by Fapesp, Brazil, and MECD, Spain (projects PB98-0932-C02-01, BFM2000-0962- C02-02 and PHB2001-0052-PC) . 1