Invariant graphs in Julia sets and decompositions of rational maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909400493457408 |
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| author | Cui, Guizhen Gao, Yan Zeng, Jinsong |
| author_facet | Cui, Guizhen Gao, Yan Zeng, Jinsong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12371 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant graphs in Julia sets and decompositions of rational maps Cui, Guizhen Gao, Yan Zeng, Jinsong Dynamical Systems 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. |
| title | Invariant graphs in Julia sets and decompositions of rational maps |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2408.12371 |