On a theorem of Rees-Shishikura
This article provides a generalisation of a theorem of Rees and Shishikura see [\textit{M. Shishikura}, Lond. Math. Soc. Lect. Note Ser. 274, 289--305 (2000; Zbl 1062.37039)] which states that if a formal mating of two (postcritically finite) polynomials is equivalent to a rational map, then the topological mating is conjugate to the same rational map. More precisely, the authors show that if the equivalence is given by a pair \((\phi_0,\phi_1)\) of homeomorphisms, such that \(\phi_0\) is a local conjugacy between the branched covering \(F\) and the rational map \(f\) in a neighbourhood of the critical cycles, then the Thurston pullback sequence \(\{ \phi_n \}\) converges uniformly to the semiconjugacy. The article ends with an application, where the authors exhibit a new type of surgery called ``folding''. This construction was introduced in the authors' article [``Renormalization and wandering continua of rational maps '', Preprint, \url{arXiv:1105.2935}]. Using the main theorems of the present paper, it is shown that if a folding \(F\) is Thurston equivalent via the pair of homeomorphisms \((h_0,h_1)\) to a rational map \(R\), then this equivalence can be promoted to a semiconjugacy between the folding and the rational map. It seems that this new surgery could lead to interesting examples of rational maps.
posctcritically finite branched coverings
scientific article; zbMATH DE number 6167098
scientific article