  In this paper we study the geometry of the attractors of holomorphic maps
with an irrationally indifferent fixed point. We prove that for an open set of
such holomorphic systems, the local attractor at the fixed point has Hausdorff
dimension two, provided the asymptotic rotation at the fixed point is of
sufficiently high type and does not belong to Herman numbers. As an immediate
corollary, the Hausdorff dimension of the Julia set of any such rational map
with a Cremer fixed point is equal to two. Moreover, we show that for a class
of asymptotic rotation numbers, the attractor satisfies Karpi\'{n}ska's
dimension paradox. That is, the the set of end points of the attractor has
dimension two, but without those end points, the dimension drops to one.
