Iterated images and the plane Jacobian conjecture

Pages: 455 - 461, Volume 16, Issue 2, October 2006

doi:10.3934/dcds.2006.16.455       Abstract        Full Text (185.7K)       Related Articles

Ronen Peretz - Dept. of Math., Ben Gurion University of the Negev, Beer-Sheva, 84105, Israel (email)
Nguyen Van Chau - Institute of Mathematics, P.O. Box 1078, Hanoi, Vietnam (email)
L. Andrew Campbell - 908 Fire Dance Lane, Palm Desert CA 92211, United States (email)
Carlos Gutierrez - ICMC-USP, São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil (email)

Abstract: We show that the iterated images of a Jacobian pair $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ stabilize; that is, all the sets $f^k(\mathbb{C}^2)$ are equal for $k$ sufficiently large. More generally, let $X$ be a closed algebraic subset of $\mathbb{C}^N$, and let $f:X\rightarrow X$ 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 $\Omega \subseteq X$ with $f(\Omega) \subseteq \Omega$, the sets $f^k(\Omega)$ stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

Keywords:  Stable image, polynomial map, etale, Jacobian conjecture.
Mathematics Subject Classification:  Primary: 14R15; Secondary: 14E09, 14E07.

Received: May 2005;      Revised: August 2005;      Published: July 2006.