Invariant Peano curves of expanding Thurston maps

Abstract

We consider Thurston maps, i.e., branched covering maps f:S² → S² 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 = fⁿ of f is semi-conjugate to z^d: S¹ → S¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ: S¹ → S² (onto), such that F∘γ(z) = γ(z^d) (for all z∈S¹).