Correction and linearization of resonant vector fields and diffeomorphisms

We extend the classical Siegel-Brjuno-Rüssmann linearization theorem to the resonant case by showing that under A. D. Brjuno's diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property that, when substracted from the vector field (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 spite of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by pure combinatorics, without any analysis.