Properties of orbit type diagrams

For fun, I created a 3d gravity simulation and colored each $(x,y)$ coordinate blue or green if a light mass put at that location would collide – within some number of simulation steps – with one of the masses of a binary system of two heavy masses, with its center of mass at the center of the image.

The result looks very fractal and in some way chaotic.

In this paper by J. Nagler, these are referred to as "orbit type diagrams" (OTDs).

Is it possible to compute such a map without having to simulate the orbits (analytical solution)?

What can be said about the sensitivity of the trajectory on the initial conditions?

What happens in more complex systems?

If relativistic effects are considered, is the result still fractal?