On the number of topological orbits of complex germs in classes (xy, xa + yb)

Abstract

We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x2 + y5) has two topological orbits.