On a theorem of Rees-Shishikura. (English. French summary) Zbl 1283.37050

Cui, Guizhen; Peng, Wenjuan; Tan, Lei

Ann. Fac. Sci. Toulouse, Math. (6) 21, No. 5, Spec. Issue, 981-993 (2012).

This article provides a generalisation of a theorem of Rees and Shishikura see [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, 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.

Reviewer: Tom Sharland (Stony Brook)

MSC:

37F20	Combinatorics and topology in relation with holomorphic dynamical systems
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Keywords:

Thurston equivalence; holomorphic dynamics; posctcritically finite branched coverings

Citations:

Zbl 1062.37039

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