A DECOMPOSITION THEOREM FOR HERMAN MAPS

Abstract. In 1980s, Thurston established a topological charac-terization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed ‘Herman map ’ is developed. It’s shown that every Herman map can be decomposed along a stable multic-urve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result gives an answer to a problem of McMullen in a sense and enables us to prove a Thurston-type theorem for rational maps with Herman rings. 1