$C^{1}$-Stable-Manifolds for Periodic Heteroclinic Chains in Bianchi IX: Symbolic Computations and Statistical Properties

In this paper we study oscillatory Bianchi models of class A and are able to show that for admissible periodic heteroclinic chains in Bianchi IX there exisist $C^{1}$- stable - manifolds of orbits that follow these chains towards the big bang. A detailed study of Takens Linearization Theorem and the Non-Resonance-Conditions leads us to this new result in Bianchi class A. More precisely, we can show that there are no heteroclinic chains in Bianchi IX with constant continued fraction development that allow Takens-Linearization at all of their base points. Geometrically speaking, this excludes "symmetric" heteroclinic chains with the same number of "bounces" near all of the 3 Taub Points - the result shows that we have to require some "asymmetry" in the bounces in order to allow for Takens Linearization, e.g. by considering admissible 2-periodic continued fraction developments. We conclude by discussing the statistical properties of those solutions, including their topological and measure-theoretic genericity.