Stability and infinitesimal stability

Abstract

In this Chapter a linearisation method is described for determining whether a given differentiable map-germ is stable. The gist of the method consists in reducing the question to the linear problem of infinitesimal stability and to the practically more easily solved problem of infinitesimal V-stability. We develop the technique necessary for the foundation of the method and apply it to the simplest situation, proving a theorem about the equivalence of a function to its Taylor polynomial in a neighbourhood of a critical point of finite multiplicity.