Iterated images and the plane Jacobian conjecture

Ronen Peretz; Nguyen Van Chau; L. Andrew Campbell; Carlos Gutierrez

doi:10.3934/dcds.2006.16.455

Iterated images and the plane Jacobian conjecture

Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez

Department of Mathematics

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

2 Scopus citations

Abstract

We show that the iterated images of a Jacobian pair f : ℂ² → ℂ² stabilize; that is, all the sets f^k(ℂ ²) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of ℂ^N, and let f : X → X be an open polynomial map with X - f(X) a finite set. We show that the sets f ^k(X) stabilize, and for any cofinite subset Ω ⊆ X with f(Ω) ⊆ Ω, the sets f^k(Ω) stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

Original language	English
Pages (from-to)	455-461
Number of pages	7
Journal	Discrete and Continuous Dynamical Systems
Volume	16
Issue number	2 SPEC. ISS.
DOIs	https://doi.org/10.3934/dcds.2006.16.455
State	Published - 1 Jan 2006

Keywords

Jacobian conjecture
Polynomial map
Stable image
Étale

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2006.16.455

Cite this

@article{dad60c685a13478ea5561d88f2e44d01,

title = "Iterated images and the plane Jacobian conjecture",

abstract = "We show that the iterated images of a Jacobian pair f : ℂ2 → ℂ2 stabilize; that is, all the sets fk(ℂ 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of ℂN, and let f : X → X be an open polynomial map with X - f(X) a finite set. We show that the sets f k(X) stabilize, and for any cofinite subset Ω ⊆ X with f(Ω) ⊆ Ω, the sets fk(Ω) stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.",

keywords = "Jacobian conjecture, Polynomial map, Stable image, {\'E}tale",

author = "Ronen Peretz and {Van Chau}, Nguyen and {Andrew Campbell}, L. and Carlos Gutierrez",

year = "2006",

month = jan,

day = "1",

doi = "10.3934/dcds.2006.16.455",

language = "English",

volume = "16",

pages = "455--461",

journal = "Discrete and Continuous Dynamical Systems",

issn = "1078-0947",

publisher = "American Institute of Mathematical Sciences",

number = "2 SPEC. ISS.",

}

TY - JOUR

T1 - Iterated images and the plane Jacobian conjecture

AU - Peretz, Ronen

AU - Van Chau, Nguyen

AU - Andrew Campbell, L.

AU - Gutierrez, Carlos

PY - 2006/1/1

Y1 - 2006/1/1

N2 - We show that the iterated images of a Jacobian pair f : ℂ2 → ℂ2 stabilize; that is, all the sets fk(ℂ 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of ℂN, and let f : X → X be an open polynomial map with X - f(X) a finite set. We show that the sets f k(X) stabilize, and for any cofinite subset Ω ⊆ X with f(Ω) ⊆ Ω, the sets fk(Ω) stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

AB - We show that the iterated images of a Jacobian pair f : ℂ2 → ℂ2 stabilize; that is, all the sets fk(ℂ 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of ℂN, and let f : X → X be an open polynomial map with X - f(X) a finite set. We show that the sets f k(X) stabilize, and for any cofinite subset Ω ⊆ X with f(Ω) ⊆ Ω, the sets fk(Ω) stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

KW - Jacobian conjecture

KW - Polynomial map

KW - Stable image

KW - Étale

UR - http://www.scopus.com/inward/record.url?scp=33748525204&partnerID=8YFLogxK

U2 - 10.3934/dcds.2006.16.455

DO - 10.3934/dcds.2006.16.455

M3 - Article

AN - SCOPUS:33748525204

SN - 1078-0947

VL - 16

SP - 455

EP - 461

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 2 SPEC. ISS.

ER -

Iterated images and the plane Jacobian conjecture

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this