Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes

TL;DR

This paper demonstrates the existence of Bianchi spacetimes exhibiting aperiodic oscillatory behavior near singularities, supporting the conjecture that their dynamics are driven by the Kasner map, a circle transformation.

Contribution

It proves that certain non-periodic Kasner map orbits attract solutions of the Bianchi spacetime equations, establishing aperiodic asymptotic behavior.

Findings

Existence of Bianchi spacetimes with aperiodic oscillations

Attraction of non-periodic Kasner orbits to spacetime solutions

Support for the Kasner map conjecture in Bianchi dynamics

Abstract

We study the asymptotic behavior of type A vacuum Bianchi spacetimes. It has been conjectured that this behavior is driven by a certain circle map, called the Kasner map. As a step towards 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 type A vacuum Bianchi spacetimes. The orbits of the Kasner map for which we can prove such a result are those which are not periodic and which do not accumulate on any periodic orbit. This shows the existence of Bianchi spacetimes with a aperiodic oscillatory asymptotic behavior.