Invariant Jordan curves of Sierpiński carpet rational maps

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 $R$.