Simultaneous linearization of germs of commuting holomorphic diffeomorphisms

Let alpha(l), ... , alpha(n) be irrational numbers which are linearly independent over the rationals, and f(l), ... , f(n), commuting germs of holomorphic diffeomorphisms in C such that f(k)(0) = 0, f(k)'(0) = e(2 pi i alpha k), k = 1, ... , n. Moser showed that fl, ... , f(n) are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations R-alpha k (z) = e(2 pi i alpha k) z) if the vector of rotation numbers (alpha(l), ... , alpha(n)) satisfies a Diophantine condition. Adapting Yoccoz's renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition B(alpha(l), ... , alpha(n)) < +infinity, where B(alpha(l), ... , alpha(n)) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f(l), ... , f(n) in a neighbourhood of the origin, then a weaker arithmetic condition B'(alpha(l), ... , alpha(n)) < +infinity analogous to Perez-Marco's condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii Ce-2 pi B, Ce-2 pi B' for some universal constant C.