Hi Ahmed,

I don't have any resources that I can think of. The general theory may or may not be connected to ergodic theory (not my field). I understand that one question ergodic theory deals with is whether the trajectory of a system will pass arbitrarily close to any given state/position. For instance, if I remember correctly, once a spacecraft has escaped Earth (sufficiently), then the 3-body system Sun-Jupiter-spacecraft is "ergodic": that is, there is a trajectory from the Earth to any point in the solar system if you're willing to wait long enough. (The is different that the general 3-body problem in that the gravitational effects of the mass of the spacecraft on the Sun and Jupiter are neglected.) I believe they use it to plan space missions. I'm not sure how one goes about finding the desired path though. As I recall, it was connected to symbolic dynamics.

In the above system, I suppose one would note that the $z$ coordinate changes monotonically along a path. So to state the obvious, you can't get to a lower position from a higher position by going up. Consequently, the passes between the peaks play an important on the upper part of the surface. Aside from such common-sense observation, I don't have anything to add.

Cheers!