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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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