Invariant graphs in Julia sets and decompositions of rational maps

Guizhen Cui; Yan Gao; Jinsong Zeng

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arXiv · 2408.12371

Invariant graphs in Julia sets and decompositions of rational maps

Guizhen Cui·Yan Gao·Jinsong Zeng

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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.

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BibTeX RIS

Guizhen Cui, Yan Gao, Jinsong Zeng. 2024-08-22. Invariant graphs in Julia sets and decompositions of rational maps. https://arxiv.org/abs/2408.12371

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