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Iterated images and the plane Jacobian conjecture

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  • 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.
    Mathematics Subject Classification: Primary: 14R15; Secondary: 14E09, 14E07.


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