Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps f:S ² → S ² that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficiently high iterate F = f ⁿ of f is semi-conjugate to z ^d :S ¹ → S ¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ:S ¹ → S ² (onto), such that F°γ(z) = γ(z ^d ) (for all z∈S ¹).