Local equivalence and conjugacy of families of vector fields and diffeomorphisms

Abstract

In this thesis we work with C∞ or analytic families of vector fields or diffeomorphisms. We are interested in local equivalences and conjugacies between such families and families in a “simple” form, sometimes called a normal form. Traditionally this normal form is chosen to be linear, but in some cases this choice prevents us from obtaining an analytic equivalence or conjugacy. So in such cases we will allow the presence of non-linear terms in the normal form. A lot of work has already been done for individual vector fields and diffeomorphisms. It turns out that the eigenvalues of the linear part of the vector field, resp. diffeomorphism at the singular, resp. fixed point are determining whether the vector field or diffeomorphism is equivalent with or conjugate to its linear part. If the eigenvalues form a hyperbolic non-resonant set then there are celebrated results from Poincaré and Siegel telling us when an analytic conjugacy with the linear part can be obtained. If the eigenvalues form a hyperbolic resonant set then sometimes it is possible to obtain a finitely smooth conjugacy. In the non-hyperbolic case it becomes much more difficult to obtain smooth equivalences and conjugacies. As in this thesis we are working with families of vector fields or diffeomorphisms, we will encounter the same problems concerning hyperbolicity and resonance as in the case of individual systems. An additional problem can be caused by the parameters that are in play in a family. As the parameter perturbs the eigenvalues, it can cause resonances which are absent for the unperturbed system. This phenomenon also has its impact on the smoothness of the equivalence or conjugacy.