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Invariant Jordan curves of Sierpiński carpet rational maps. (English) Zbl 1441.37051

Summary: In this paper, we prove that if \(R:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}\) is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer \(n_{0}\), such that, for any \(n\geq n_{0}\), there exists an \(R^{n}\)-invariant Jordan curve \(\Gamma\) containing the postcritical set of \(R\).

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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