We consider Thurston maps, i.e., branched covering maps f:S2→S2
that are postcritically finite. In addition, we assume that f is expanding in
a suitable sense. It is shown that each sufficiently high iterate F=fn of
f is semi-conjugate to zd:S1→S1, where d is equal to the
degree of F. More precisely, for such an F we construct a Peano curve
γ:S1→S2 (onto), such that F∘γ(z)=γ(zd)
(for all z∈S1).Comment: 63 pages, 12 figure