Matteo Ruggiero – författare

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X, x0) --> (X, x0), where X is a complex surface having x0 as a normal singularity. We prove that as long as x0 is not a cusp singularity of X, then it is possible to find arbitrarily high modifications ?: X? --> (X, x0) such that the dynamics of f (or more precisely of fN for N big enough) on X? is algebraically stable. This result is proved by understanding the dynamics induced by f on a space of valuations associated to X; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.

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.