The dynamics of holomorphic germs near a curve of fixed points

One of the interesting areas in the study of the local dynamics in several complex variables is the dynamics near the origin OO of maps tangent to the identity, that is of germs of holomorphic self-maps f:n→n such that f(O)=Of(O)=O and dfO=id. When n=1n=1 the dynamics is described by the known Leau–Fatou flower theorem but when n>1n>1, we are still far from understanding the complete picture, even though very important results have been obtained in recent years (see, e.g., [2], [7], [10] and [19]). In this note we want to investigate conditions ensuring the existence of parabolic curves (the two-variable analogue of the petals in the Leau–Fatou flower theorem) for maps tangent to the identity in dimension 2. Using simple examples, we prove that these conditions are not, generally, sufficient.