The Dynamical Behaviour of some Automorphisms of C^2 that fixes the axes

Abstract

We use existing theory of Hakim and Weickert in order to identify a complex line which is such that close to the origin the stable set of the origin will be situated close to this line. Then we study the dynamical behaviour of an approximation to these automorphisms close to the origin. We prove that for this approximation points that are sufficiently close to the origin and one of the complex axes will eventually converge under iteration to this complex axis. On the complex line we show that points will approach the origin. Points that are close to, but not on, the line where one of the coordinates approach zero will also eventually hit one of the complex axes.