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Invariant Jordan curves of Sierpiński carpet rational maps

Published online by Cambridge University Press:  09 September 2016

YAN GAO
Affiliation:
Mathematical School of Sichuan University, 610065, PR China email gyan@scu.edu.cn
PETER HAÏSSINSKY
Affiliation:
Université d’Aix-Marseille, Institut de Mathématiques de Marseille (I2M), 39, rue Frédéric Joliot Curie 13453, Marseille Cedex 13, France email phaissin@math.univ-toulouse.fr
DANIEL MEYER
Affiliation:
Department of Mathematics and Statistics, PO Box 35, FI-40014 University of Jyväskylä, Finland email dmeyermail@gmail.com
JINSONG ZENG
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China email zeng.jinsong@amss.ac.cn

Abstract

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 $\unicode[STIX]{x1D6E4}$ containing the postcritical set of $R$.

Type
Original Article
Copyright
© Cambridge University Press, 2016 

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