In a previous post I wrote about the Mixmaster model. I mentioned that not much had happened in this area for many years and that I was involved in an attempt to change this. I was happy to see that two papers have recently appeared on the arXiv which show that this area of research is getting moving again. To describe the results of these papers I first need some background. The Wainwright-Hsu system is a system of ODE which describes solutions of the Einstein equations of several Bianchi types, including types IX (which is Mixmaster) and I. The -limit set of a solution of type IX contains points of type I and it is interesting to know what subsets of the set of all vacuum type I solutions (the Kasner circle ) can occur as -limit sets. Apart from three exceptional points (the Taub points) each point of the Kasner circle is the -limit set of a unique solution of Bianchi type II. The -limit set of that solution of Bianchi type II is also a point of the Kasner circle, call it . This defines a mapping from the complement of the Taub points in to , the Kasner mapping. It is suggestive (and widely assumed in the physics literature) that the dynamics of Bianchi IX solutions at early times should be approximated by the discrete dynamics defined by the Kasner map. On a rigorous level essentially nothing was known about this.
The simplest orbit of the Kasner mapping is an orbit of period three which is unique up to symmetry transformations. In a paper by Marc Georgi, Jörg Härterich, Stefan Liebscher and Kevin Webster (arXiv:1004.1989) they studied this orbit. The three points in this orbit are joined by three orbits of type II, forming a heteroclinic cycle. The question is if there are solutions of Bianchi type IX which have this heteroclinic orbit as -limit set and if so how many. In the paper just mentioned the authors show that there is a Lipschitz submanifold invariant under the Bianchi IX flow with the properties that all solutions starting at points on this manifold converge to the heteroclinic cycle in the past time direction. They do this for the Einstein vacuum equations and for the Einstein equations coupled to a perfect fluid with linear equation of state. If the equation of state in the latter case is then the result is restricted to the range . They also indicate various possible generalizations of this result. The method of proof is a careful study of a return map which describes the behaviour of solutions starting near the heteroclinic cycle. In the end it is shown that this return map defines a contraction on a space of functions whose graphs are candidates for the invariant manifold of interest.
The other paper by François Béguin (arXiv:1004.2984) concentrates on orbits of the Kasner map which, while keeping away from the Taub points, are otherwise as general as possible. The reason for the genericity assumption is to avoid resonances. To put this into context I recall the following facts. Let be a stationary solution of a dynamical system. Then the system can be linearized about that point. It can be asked under what circumstances the system can be transformed into its linearization by a homeomorphism, in a local neighbourhood of . The Hartman-Grobman theorem says that this is the case provided no eigenvalue of the linearization is purely imaginary. The corresponding statement for a diffeomorphism of class is not true. There is an analogue of the Hartman-Grobman theorem in the case, due to Sternberg, but it requires extra assumptions. These assumptions are conditions on linear combinations of eigenvalues with integer coefficients. The assumptions of Béguin are related to these conditions. In the Mixmaster problem there is an extra complication. The presence of the Kasner circle means that the linearization about any point automatically has a zero eigenvalue. Thus what is needed is an analogue of Sternberg’s theorem in certain circumstances where a zero eigenvalue is present. There is a theorem of this type, due to Takens. I heard about it for the first term when Béguin gave a talk on this subject in out institute. In any case, Takens’ theorem is at the heart of the proof of the paper. It allows certain things to be linearized in a way which is differentiable.