Invariant graphs in Julia sets and decompositions of rational maps

TL;DR

This paper demonstrates the existence of invariant graphs within Julia sets of post-critically finite rational maps, providing a new decomposition approach based on cluster-Sierpinski structures.

Contribution

It introduces a novel method to construct finite, connected invariant graphs in Julia sets that contain all post-critical points, utilizing cluster-Sierpinski decomposition.

Findings

Existence of invariant graphs for large iterates of post-critically finite rational maps.

Invariant graphs contain all post-critical points in the Julia set.

Decomposition of the Riemann sphere into regions with at most one post-critical point each.

Abstract

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$, and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$ such that $f^n(G) \subset G$. This graph contains all post-critical points in the Julia set, while every component of $\overline{\mathbb{C}}\setminus G$ contains at most one post-critical point in the Fatou set. The proof relies on the cluster-Sierpinski decomposition of post-critically finite rational maps.