In the framework of mathematical cosmology, Bianchi IX model has great importance due to its stochastic properties. I'm reading a publication in which is claimed
The use of the invariant measure and of the Artin's theorem provides the complete equivalence between the BKL piece-wise description and the Misner-Chitré continuous one.
my problem here is: what does Artin's theorem state? In references the following work is cited
E. Artin, "Ein Mechanisches System mit quasi-ergodischen Bahnen, Collected papers" (1965)
Unfortunately, I can't read deutch (all I can catch is that is a work on the ergodic theory of a certain mechanical system) and I haven't found the reference, really.
So, does anyone know which is the statement of Artin theorem? (Or can indicate a reference?)
Remark. I know the question arises in framework of mathematical physics, but I'm concerned only with a result that should be a mathematical one.
Artin's paper describes the symbolic dynamics of geodesics in the upper half plane model of hyperbolic geometry. He chooses the fundamental domain of the modular surface and reflects geodesics back onto the surface using a continued fraction map. His statement is that out of all geodesics that pass through a given point of the surface , almost all are quasi-ergodic .
There are many recent papers that use this construction and the notion of quasi-ergodic has been restated or refined in these later papers . The original idea is due to Artin.
It should be noted that this is a quotient space of Hyperbolic space. I don't remember what the Bianchi IX model describes.