This thesis deals with specific featuresof the theory of holomorphic dynamics in dimension 2 and then sets out to studyanalogous questions in higher dimensions, e.g. dealing with normal forms forrigid germs, and examples of Kato 3-folds.

The local dynamics of holomorphic mapsaround critical points is still not completely understood, in dimension 2 orhigher, due to the richness of the geometry of the critical set for alliterates.

In dimension 2, the study of thedynamics induced on a suitable functional space (the valuative tree) allows aclassification of such maps up to birational conjugacy, reducing the problem tothe special class of rigid germs, where the geometry of the critical set issimple.

In some cases, from such dynamical dataone can construct special compact complex surfaces, called Kato surfaces,related to some conjectures in complex geometry.