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- W4313447568 abstract "An orientation-preserving branched covering map $fcolon S^2 to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between Mobius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map $f$ is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair $(G,varphi)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $varphi$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $G$. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair $(G,varphi)$ associated with a critically fixed Thurston map $f$. Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs $(G, mathrm{id}_{S^2})$." @default.
- W4313447568 created "2023-01-06" @default.
- W4313447568 creator A5039016571 @default.
- W4313447568 creator A5049270375 @default.
- W4313447568 date "2022-12-30" @default.
- W4313447568 modified "2023-09-26" @default.
- W4313447568 title "Critically fixed Thurston maps: classification, recognition, and twisting" @default.
- W4313447568 doi "https://doi.org/10.48550/arxiv.2212.14759" @default.
- W4313447568 hasPublicationYear "2022" @default.
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