A model for planar compacta and rational Julia sets
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913466065879040 |
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| author | Luo, Jun Yang, Yi Yao, Xiaoting |
| author_facet | Luo, Jun Yang, Yi Yao, Xiaoting |
| contents | A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that there is a finest upper semi-continuous decomposition of K into subcontinua such that the resulting quotient space is a Peano compactum. We call this decomposition the core decomposition of K with Peano quotient and its elements atoms of K. We show that for any branched covering f of the extended complex plane onto itself and for any atom d of K, the preimage of d under f has finitely many components each of which is an atom of the preimage of K under f. Since rational functions are branched coverings, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial and K completely invariant under f. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_03773 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A model for planar compacta and rational Julia sets Luo, Jun Yang, Yi Yao, Xiaoting Dynamical Systems 2020: 37B45, 37F10, 54D05 A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that there is a finest upper semi-continuous decomposition of K into subcontinua such that the resulting quotient space is a Peano compactum. We call this decomposition the core decomposition of K with Peano quotient and its elements atoms of K. We show that for any branched covering f of the extended complex plane onto itself and for any atom d of K, the preimage of d under f has finitely many components each of which is an atom of the preimage of K under f. Since rational functions are branched coverings, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial and K completely invariant under f. |
| title | A model for planar compacta and rational Julia sets |
| topic | Dynamical Systems 2020: 37B45, 37F10, 54D05 |
| url | https://arxiv.org/abs/2209.03773 |