I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential topology, general relativity, Hamilton–Jacobi equation, and see why in integrable systems the motion takes place on an N torus and in general fills the surface. Also, know that even the reduced three-body problem could have a chaotic, highly irregular dynamical evolution.
Referring to what is reported in the news you can see that specifically one of her discoveries show that even the potentially chaotic dynamic of the three-body problem follows some deeply geometric laws. What we consider unpredictable, in the interaction of Sun, Moon, and Earth, is still to some extent predictable.
Is it possible for you to explain this restriction on phase space using what I now know and understand, I mean without moduli spaces and ergodic theory?
In addition I greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory assuming understanding of the essential topology used in general relativity.