CORRECTION AND LINEARIZATION OF RESONANT VECTOR-FIELDS AND DIFFEOMORPHISMS

We extend the classical Siegel-Brjuno-Russmann Linearization theorem t o the resonant case by showing that under A. D. Brjuno's diophantine c ondition, any resonant local analytic vector field (resp. diffeomorphi sm) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant te rms and (3) has the property that, when substracted from the vector fi eld (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism thus ''corrected'' becomes analytically linearizable (with a privileged or ''canonical'' Linearizing map). Moreover, in sp ite of the small denominators and contrary to a hitherto prevalent opi nion, the correction's analyticity can be established by pure combinat orics, without any analysis.