Claes Uggla: Oscillatory spacelike singularities: The Bianchi type VI_{-1/9} vacuum models.

The Bianchi type \(\mathrm{VI}_{-1/9}\), \(\mathrm{VIII}\) and \(\mathrm{IX}\) vacuum models all have 4-dimensional Hubble-normalized state spaces and all are expected to have generic initial oscillatory singularities, but, arguably, it is the type \(\mathrm{VI}_{-1/9}\) models rather than the type \(\mathrm{VIII}\) and \(\mathrm{IX}\) models that are the next natural step toward a better understanding of generic spacelike singularities. Unfortunately, the type \(\mathrm{VI}_{-1/9}\) models constitute a more formidable challenge than the type \(\mathrm{VIII}\) and \(\mathrm{IX}\) models since the invariant boundary sets responsible for the oscillations are much more complicated than those for type \(\mathrm{VIII}\) and \(\mathrm{IX}\). For the first time, the equations on these type \(\mathrm{VI}_{-1/9}\) boundary sets are explicitly solved. In addition, a new graphic representation is introduced, which gives a complete description of the associated network of heteroclinic chains. In particular, this is illustrated with examples of networks of entangled cyclic heteroclinic chains for which it is shown that only some of the cyclic chains are asymptotically relevant.