Iterated Images and the Plane Jacobian Conjecture Academic Article uri icon

abstract

  • We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More generally, we obtain the same result for open polynomial maps of a closed algebraic subset X of complex N-space to itself that have finite coimage, and for cofinite subsets of such an X invariant under the map. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity. Comment: 7 pages, AMSLaTeX, revised in response to referees' suggestions, typos corrected, refrences expanded, to appear in Discrete and Continuous Dynamical Systems

publication date

  • January 1, 2006