A Characterization of Semisimple Plane Polynomial Automorphisms
  It is well-known that an element of the linear group ${\rm GL}_n(\C)$ is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms. 
Jean-Philippe Furter, Stefan Maubach