Rigid Germs, the Valuative Tree, and Applications to Kato Varieties

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

The local dynamics of holomorphic maps
around critical points is still not completely understood, in dimension 2 or
higher, due to the richness of the geometry of the critical set for all
iterates.

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









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