Thurston-type Theorem For McMullen Mapping Dynamical Systems To Rational Mappings With Rotational Fields

The thesis mainly consists of two subjects:·Dynamics of McMullen maps. In this part, we study the local connectivity of Julia sets for rational maps. We develop Yoccoz puzzle techniques to study McMullen maps and show that the boundary of the basin of infinity is always a Jordan curve if the Julia set is not a Cantor set. This give a positive answer to Devaney's question. we also show the Julia set of McMullen maps is locally connected except some special cases.·Thurston's theory on characterization of rational maps and extensions.For this subject, we establish a'Decomposition Theorem':Every non-parabolic branched covering can be decomposed along a stable multicurve into finitely many Siegel maps or Thurston maps, such that the combinatorics and rational realizations of these resulting maps essentially dominate the original one.These resulting maps can be considered as the renormalizations of the original map. The motivation to establish such a theorem is to prove a Thurston-type theorem for rational maps with Herman rings. The Decomposition Theorem implies:Thurston-type theorems for rational maps with Herman rings can be reduced to Thurston-type theorems for rational maps with Siegel disks.The Decomposition Theorem allows us to extend Thurston's Theorem to many poscritically infinite cases and give characterizations of rational maps with attracting cycles, Siegel disks and Herman rings. On the other hand, it allows us to construct many branched coverings without Thurston obstructions but not equivalent to ratio-nal maps...