# MathSciDoc: An Archive for Mathematician ∫

#### Classical Analysis and ODEsComplex Variables and Complex Analysismathscidoc:1701.05001

Acta Mathematica, 210, (1), 95-171, 2010.11
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}).
Expanding Thurston map; Invariant Peano curve
```@inproceedings{daniel2010invariant,
title={Invariant Peano curves of expanding Thurston maps},
author={Daniel Meyer},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400322677767},
booktitle={Acta Mathematica},
volume={210},
number={1},
pages={95-171},
year={2010},
}
```
Daniel Meyer. Invariant Peano curves of expanding Thurston maps. 2010. Vol. 210. In Acta Mathematica. pp.95-171. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203400322677767.