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Invariant Peano curves of expanding Thurston maps. (English) Zbl 1333.37043
Summary: We consider Thurston maps, i.e., branched covering maps $$f:S^2 \to S^2$$ 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 $$z^d:S^1 \to S^1$$, where $$d = \deg F$$. More precisely, for such an $$F$$ we construct a Peano curve $$\gamma:S^1 \to S^2$$ (onto), such that $$F \circ \gamma(z) = \gamma(z^d)$$ (for all $$z \in S^1$$).

##### MSC:
 37F20 Combinatorics and topology in relation with holomorphic dynamical systems 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 57M12 Low-dimensional topology of special (e.g., branched) coverings 57M50 General geometric structures on low-dimensional manifolds 28A80 Fractals
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