On the postcritical set of a rational map

Laura G. DeMarco; Sarah C. Koch; Curtis T. McMullen

doi:10.1007/s00208-018-1732-6

On the postcritical set of a rational map

Laura G. DeMarco^*, Sarah C. Koch, Curtis T. McMullen

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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)
Pages (from-to)	1-18
Number of pages	18
Journal	Mathematische Annalen
Volume	377
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-018-1732-6
State	Published - Jun 1 2020

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Mathematics(all)

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title = "On the postcritical set of a rational map",

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{\textquoteright}s theorem and iteration on Teichm{\"u}ller space.",

author = "DeMarco, {Laura G.} and Koch, {Sarah C.} and McMullen, {Curtis T.}",

note = "Publisher Copyright: {\textcopyright} 2018, Springer-Verlag GmbH Germany, part of Springer Nature.",

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AU - DeMarco, Laura G.

AU - Koch, Sarah C.

AU - McMullen, Curtis T.

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N2 - 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.

AB - 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.

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On the postcritical set of a rational map

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