COMBINING RATIONAL MAPS AND CONTROLLING OBSTRUCTIONS

We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combi nation theorems for postcritically finite rational maps. Our construct ions increase the degree of the map but always yield branched covering s which are equivalent to rational maps, independent of the combinator ics of the original map. The main tool is a general theorem based on t he intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.