The Mixmaster cosmological metrics

This paper begins with a short presentation of the Bianchi IX or ``Mixmaster'' cosmological model, and some ways of writing the Einstein equations for it. There is then an interlude describing how I came to a study of this model, and then a report of some mostly unpublished work from a Ph.\ D. thesis of D. M. (Prakash) Chitre relating approximate solutions to geodesic flows on finite volume negative curvature Riemannian manifolds, for which he could quote results on ergodicity. A final section restates studies of a zero measure set of solutions which in first approximation appear to have only a finite number of Kasner epochs before reaching the singularity. One finds no plausible case for such behavior in better approximations.