Invariant Peano curves of expanding Thurston maps

Details Invariant Peano curves of expanding Thurston maps Invariant Peano curves of expanding Thurston maps

2009-07-09

arxiv.org/abs/0907.1536v3

Mathematics

Complex Variables

62 pages, 12 figures

Scientific paper

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically 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 $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).

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