Holomorphic motions and structural stability of polynomial automorphisms of \mathbf{C}^{2} Gregery BuzzardMichael Jenkins 37F1037F1534D30structural stabilityHenon maphyperbolicholomorphic motion 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.) Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3252 10.1512/iumj.2008.57.3252 en Indiana Univ. Math. J. 57 (2008) 277 - 308 state-of-the-art mathematics http://iumj.org/access/