One of the remarkable differences between complex analysis in one and
several complex variables is contained in the rigidity phenomenon for
mappings between real submanifolds in multi-dimensional complex space. A
good illustration of the rigidity is the fact that any formal invertible
power series mapping between nondegenerate real hypersurfaces in complex
space is always a holomorphic mapping.
In this talk we establish a connection between real hypersurfaces with
degeneracies and singular second order complex differential equations. We
then demonstrate how the local holomorphic dynamics of the singular ODEs
enables one to construct counterexamples to the above convergence
phenomenon.