Invariant Jordan curves of Sierpiski carpet rational maps

TL;DR

This paper proves the existence of invariant Jordan curves containing the postcritical set for certain postcritically finite rational maps with Julia sets homeomorphic to the Sierpiński carpet, revealing structural properties of these complex dynamical systems.

Contribution

It establishes the existence of invariant Jordan curves for iterates of postcritically finite rational maps with Sierpiński carpet Julia sets, a new structural insight.

Findings

Existence of invariant Jordan curves for large iterates of the map

Invariant curves contain the postcritical set

Results apply to maps with Julia set homeomorphic to the Sierpiński carpet

Abstract

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.