Critically finite maps, attractors and local dynamics.

摘要

In this thesis, we study three closely related topics: critically finite maps on Pk, attractors on Pk and local dynamics of quasi-parabolic diffeomorphisms in Cn.; First, we discuss the properties of critically finite maps on Pk based on the study of the post-critical structure. We completely classify critically finite maps on P 2. We also study two examples of critically finite maps in details, one of which is needed in the study of attractors.; Second, we study perturbations of certain regular maps on P k. We show the existence of non-algebraic chaotic attractors and study measures supported on them. For this purpose, we study systematically the properties of history spaces.; Finally, we study local holomorphic diffeomorphisms in C n with a fixed point, which are quasi-parabolic. Two problems are discussed: the holomorphic linearizability of certain germs and the existence of parabolic curves in certain cases.