The postcritical set P ( f ) of a rational map $$f:{\mathbb P}^1\rightarrow {\mathbb P}^1$$ f : P 1 → P 1 is the smallest forward invariant subset of $${\mathbb P}^1$$ P 1 that contains the critical values of f . In this paper we show that every finite set $$X\subset {\mathbb P}^1({\overline{{\mathbb Q}}})$$ X ⊂ P 1 ( Q ¯ ) can be realized as the postcritical set of a rational map. We also show that every map $$F:X\rightarrow X$$ F : X → X defined on a finite set $$X\subset {\mathbb P}^1({\mathbb C})$$ X ⊂ P 1 ( C ) can be realized by a rational map $$f:P(f)\rightarrow P(f)$$ f : P ( f ) → P ( f ) , provided we allow small perturbations of the set X . The proofs involve Belyi’s theorem and iteration on Teichmüller space.

中文翻译：

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关于有理映射的后批判集

有理映射的后临界集 P ( f ) $$f:{\mathbb P}^1\rightarrow {\mathbb P}^1$$ f : P 1 → P 1 是 $${ 的最小前向不变子集\mathbb P}^1$$ P 1 包含 f 的临界值。在本文中，我们证明了每个有限集 $$X\subset {\mathbb P}^1({\overline{{\mathbb Q}}})$$ X ⊂ P 1 ( Q¯ ) 都可以实现为后临界一组有理地图。我们还表明，每个映射 $$F:X\rightarrow X$$ F : X → X 定义在有限集 $$X\subset {\mathbb P}^1({\mathbb C})$$ X ⊂ P 1 ( C ) 可以通过有理映射 $$f:P(f)\rightarrow P(f)$$ f : P ( f ) → P ( f ) 来实现，前提是我们允许集合 X 的小扰动。证明涉及 Belyi 定理和 Teichmüller 空间上的迭代。