Abstract
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).
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Meyer, D. Invariant Peano curves of expanding Thurston maps. Acta Math 210, 95–171 (2013). https://doi.org/10.1007/s11511-013-0091-0
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DOI: https://doi.org/10.1007/s11511-013-0091-0