Title: Holomorphic motions and structural stability of polynomial automorphisms of \mathbf{C}^{2}

Authors: Adrian Jenkins and Gregery T. Buzzard

Issue: Volume 57 (2008), Issue 1, 277-308

Abstract: Combining ideas from real dynamics on compact manifolds and complex dynamics in one variable, we prove the structural stability of hyperbolic polynomial automorphisms in $\boldsymbol{C}^{2}$. We consider families of hyperbolic polynomial automorphisms depending holomorphically on the parameter $\lambda$. This is done over a series of steps - given a family $\{ f_{\lambda} \}$, where $| \lambda |$ is sufficiently small, we construct mappings defined on a neighborhood $U$ of $J_{0}$ which conjugate $f_{0}$ and $f_{\lambda}$. Moreover, it is shown that $J$ moves holomorphically. This conjugacy is then used to construct a conjugacy between $f_{0}$ and $f_{\lambda}$ defined on a neighborhood $M$ of $J_{0}^{+} \cup J_{0}^{-}$. Finally, we extend such a mapping to construct a conjugacy on all of $\boldsymbol{C}^{2}$. (See a graphic rendition of this abstract in http://www.iumj.indiana.edu/oai/2008/57/3252/3252_abstract.xml.)