  In 1980s, Thurston established a topological characterization 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
multicurve 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.
