Analytic Conjugation, Global Attractor, and the Jacobian

It is proved that the dilation \(\lambda f\) of an analytic map \(f\) on \({\bf C}^n$\) with \(f(0)=0,f'(0)=I, |\lambda|>1\) has an analytic conjugation to its linear part \(\lambda x\) if and only if \(f\) is an analytic automorphism on \({\bf C}^n\) and \(x=0\) is a global attractor for the inverse \((\lambda f)^{-1}\). This result is used to show that the dilation of the Jacobian polynomial of 12] is analyticly conjugate to its linear part.