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We study the asymptotic behavior of type A vacuum Bianchi spacetimes. It hasbeen conjectured that this behavior is driven by a certain circle map, calledthe Kasner map. As a step towards this conjecture, we prove that some orbits ofthe Kasner map do indeed attract some solutions of the system of ODEs whichdescribes the behavior of type A vacuum Bianchi spacetimes. The orbits of theKasner map for which we can prove such a result are those which are notperiodic and which do not accumulate on any periodic orbit. This shows theexistence of Bianchi spacetimes with a aperiodic oscillatory asymptoticbehavior.

Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes