Status of the three-body problem I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the topological entropy or Lyapunov exponent of the system. 
 A: Although there is no solution for the general three body problem there are many related results, especially for the restricted case.   Here is a study of LE's in restricted 3BP.  And here is a classification scheme for the mean motion resonance of the restricted 3BP including using the maximal LE for characterizing the resonances.  These will point you to many related results.  As a historical aside, Gutzwiller, in his book[1] says that the ancient Greeks knew there were 3 different lunar months (depending on how you measure them in they sky): the Sidereal month (27.32166 days), the anomalistic month (27.55455 days) and the nodical month (27.21222 days).  These result from 3 body interactions (earth, moon, sun).  The Greeks were apparently able to observe these to an accuracy of 1 second per month.  Gutzwiller calls these the first precision scientific measurements in history.  Every time I think about this I am astonished; it must have taken immense effort over centuries to compile the needed data.
[1] Martin C. Gutzwiller Chaos in Classical and Quantum systems.
