Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes

We study the asymptotic behavior of vacuum Bianchi type A spacetimes close to their singularity. It has been conjectured that this behavior is driven by a certain circle map, called the Kasner map. As a step toward this conjecture, we prove that some orbits of the Kasner map do indeed attract some solutions of the system of ODEs which describes the behavior of vacuum Bianchi type A spacetimes. The orbits of the Kasner map for which we can prove such a result are those which are not periodic and do not accumulate on any periodic orbit. This shows the existence of Bianchi spacetimes with aperiodic oscillatory asymptotic behavior.