There is this cool paper "Crash test for the restricted three-body problem" [0], that plots where the third body eventually ends up when dropped from any location. Looks very fractal-like [1][2]

[0] https://scholar.archive.org/work/wnwgyliq5fgtba45k535t5lb5e/...

[1] https://www.semanticscholar.org/paper/Crash-test-for-the-res...

[2] https://www.semanticscholar.org/paper/Crash-test-for-the-res...