Classification of critically fixed anti-Thurston maps

Lukas Geyer; Mikhail Hlushchanka

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arXiv · 2006.10788

Classification of critically fixed anti-Thurston maps

Lukas Geyer·Mikhail Hlushchanka

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Abstract

We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated to certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurstons's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. Lastly, we give some applications to the global curve attractor and twisting problems, as well as to anti-rational maps with symmetries and to critically fixed anti-polynomials.

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Lukas Geyer, Mikhail Hlushchanka. 2020-06-18. Classification of critically fixed anti-Thurston maps. https://arxiv.org/abs/2006.10788

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