Abstract
The postcritical set P(f) of a rational map f: P1→ P1 is the smallest forward invariant subset of P1 that contains the critical values of f. In this paper we show that every finite set X⊂ P1(Q¯) can be realized as the postcritical set of a rational map. We also show that every map F: X→ X defined on a finite set X⊂ P1(C) can be realized by a rational map 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.
Original language | English (US) |
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Pages (from-to) | 1-18 |
Number of pages | 18 |
Journal | Mathematische Annalen |
Volume | 377 |
Issue number | 1-2 |
DOIs | |
State | Published - Jun 1 2020 |
ASJC Scopus subject areas
- General Mathematics