Invariant graphs in Julia sets and decompositions of rational maps

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\bar{\mathbb{C}}$, 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 $\bar{\mathbb{C}}\setminus G$ 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.