I find many numerical results on the threebody 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.

3$\begingroup$ This question is too broad. There is enormous amount of knowledge about three body problem, in particular about SunMoonEarth system. One cannot describe the "status" in a short answer. $\endgroup$– Alexandre EremenkoApr 24, 2018 at 19:29
1 Answer
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.

2$\begingroup$ Exactly, @AlexandreEremenko. Don't think I mentioned instruments. Just observations which I suppose we can call measurements. Hundreds of years of observations. $\endgroup$– JohnSApr 24, 2018 at 19:35

2$\begingroup$ I am aware of the numerical estimates of the Lyapunov exponent you refer to, but i am asking for a rigorous result. $\endgroup$ Apr 24, 2018 at 20:54

$\begingroup$ Fair enough @JörgNeunhäuserer. I'll keep poking around. Good luck. $\endgroup$– JohnSApr 24, 2018 at 21:05

$\begingroup$ @JohnS By "no solution" do you mean no known bounds, in the general case, on the Lyapunov exponents or topological entropy? The reason I ask is that there is a kind of "folk belief" in certain circles that the differential equations describing the threebody problem have no solution, which is not true: projecteuclid.org/download/pdf_1/euclid.acta/1485887350 $\endgroup$ Feb 8, 2019 at 7:49