The weak stability boundary was first studied in 1987. Although it has proved important in applications with many lunar missions using low energy transfers obtained from it, its mathematical structure has remained elusive. New work surprisingly shows that this very dynamically complex region about the secondary mass point in the restricted three-body problem bounds a region analogous to a Mandelbrot set for a simple complex map. The topological structure of the boundary is determined as consisting of infinitely many Cantor sets, and gives insight on the last KAM tori about the secondary – a long sought
problem. There are interesting applications.