GLOBAL DYNAMICS OF THE MIXMASTER MODEL

The asymptotic behaviour of vacuum Bianchi models of class A near the initial singularity is studied, in an effort to confirm the standard p icture arising from heuristic and numerical approaches by mathematical proofs. It is shown that for solutions of types other than VIII and I X the singularity is velocity dominated and that the Kretschmann scala r is unbounded there, except in the explicitly known cases where the s pacetime can be smoothly extended through a Cauchy horizon. For types VIII and IX it is shown that there are at most two possibilities for t he evolution. When the first possibility is realized, and if the space time is not one of the explicitly known solutions which can be smoothl y extended through a Cauchy horizon, then there are infinitely many os cillations near the singularity and the Kretschmann scalar is unbounde d there. The second possibility remains mysterious and it is left open whether it ever occurs. It is also shown that any finite sequence of distinct points generated by iterating the Belinskii-Khalatnikov-Lifsc hitz mapping can be realized approximately by a solution of the vacuum Einstein equations of Bianchi type IX.