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