American Institute of Mathematical Sciences

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June  2006, 16(2): 455-461. doi: 10.3934/dcds.2006.16.455

Iterated images and the plane Jacobian conjecture

Ronen Peretz 1, , Nguyen Van Chau 2, , L. Andrew Campbell 3, and  Carlos Gutierrez 4,

1. 

Dept. of Math., Ben Gurion University of the Negev, Beer-Sheva, 84105, Israel
2. 

Institute of Mathematics, P.O. Box 1078, Hanoi, Vietnam
3. 

908 Fire Dance Lane, Palm Desert CA 92211, United States
4. 

ICMC-USP, São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos, SP

Received  May 2005 Revised  August 2005 Published  July 2006

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: polynomial map, etale, Jacobian conjecture., Stable image.
Mathematics Subject Classification: Primary: 14R15; Secondary: 14E09, 14E0.
Citation: Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455
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