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

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)
Journal	Mathematische Annalen
Volume	377
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-018-1732-6
State	Published - Jun 1 2020

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1007/s00208-018-1732-6

Fingerprint Dive into the research topics of 'On the postcritical set of a rational map'. Together they form a unique fingerprint.

Cite this

DeMarco, L. G., Koch, S. C., & McMullen, C. T. (2020). On the postcritical set of a rational map. Mathematische Annalen, 377(1-2). https://doi.org/10.1007/s00208-018-1732-6

@article{14a483e6c9ac4fdeb74398a6df5af2bf,

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.}",

year = "2020",

month = jun,

day = "1",

doi = "10.1007/s00208-018-1732-6",

language = "English (US)",

volume = "377",

journal = "Mathematische Annalen",

issn = "0025-5831",

publisher = "Springer New York",

number = "1-2",

}

TY - JOUR

T1 - On the postcritical set of a rational map

AU - DeMarco, Laura G.

AU - Koch, Sarah C.

AU - McMullen, Curtis T.

PY - 2020/6/1

Y1 - 2020/6/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85051517047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051517047&partnerID=8YFLogxK

U2 - 10.1007/s00208-018-1732-6

DO - 10.1007/s00208-018-1732-6

M3 - Article

AN - SCOPUS:85051517047

VL - 377

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1-2

ER -