Germs of diffeomorphisms and their Taylor expansions

Abstract

The thesis examines the relationship between the germ of a C∞ diffeomorphism f: Rn,0 -• Rn,0 which is tangent to the identity at 0 and its Taylor expansion. The case in which n is one is already well understood. For n greater than one some normal forms for germs are already known. These are germs with the property that any other germ having the same Taylor expansion is conjugate to the normal form. Conjugation may be thought of as a change of variables. The idea is that the Taylor expansion determines what the germ 'looks like'. The above concept is extended in the thesis in a new way to deal with the common situation where the Taylor expansion only partially determines what the germ 'looks like', for example the Taylor expansion may determine what the germ looks like near one axis, but not away from that axis. Examples are given. The importance of the extended concept is highlighted by a construction (using the new idea) of a large class of germs which do not have normal forms in the old, limited, sense. The theory allows one to study the centralisers of such germs, and to describe what their invariant curves 'look like', for example,’can the germs be embedded in one-parameter groups, and do they have invariant curves which may be thought of as graphs of C∞ functions