Iterated images and the plane Jacobian conjecture

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

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.

Original languageEnglish
Pages (from-to)455-461
Number of pages7
JournalDiscrete and Continuous Dynamical Systems
Volume16
Issue number2 SPEC. ISS.
DOIs
StatePublished - 1 Jan 2006

Keywords

  • Jacobian conjecture
  • Polynomial map
  • Stable image
  • Étale

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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