| Higher dimensional complex dynamics has experienced a tremendous growth in the past decade and much of this has been inspired by its connections with other fields. In Chapter II we study the periodic points that can appear for a torus action on a Kobayashi hyperbolic Stein manifold. Here we prove that under certain restrictions, there exist only a finite number of periodic points of all periods. Then we study the fixed points for an S1 action on a Stein manifold; multidimensional residue theory is used to prove the existence of at most one fixed point in many cases. We also give counterexamples of manifolds with actions having an arbitrary number of fixed points. In Chapter III, we generalize an important Corollary from Chapter II to manifolds having any finite number of generators for their second cohomology group. There are also given several applications to Siegel domains for holomorphic endomorphisms of P2; a new class of such Siegel domains is constructed.;In Chapter IV we study "the opposite case", hyperbolic maps on P2, which are proved to have no Siegel domains. We show that the set of ws-hyperbolic maps without cycles is open, which further provides a large class of examples of ws-hyperbolic: maps constructed from simpler ones. We also conjecture that the interior of the set K- of points with "bounded backward iterates", is empty, at least for certain classes of strongly hyperbolic maps on P 2. In the final section we prove that a repellor with empty interior for a special class of functions, will have Lebesgue measure zero. |
