Mixmaster: Fact and Belief

TL;DR

This paper investigates the complex dynamics of Bianchi type IX models near the initial singularity, aiming to clarify and formalize the prevailing but vague beliefs about Mixmaster behavior using dynamical systems and stochastic analysis.

Contribution

It develops a framework to sharpen and test core beliefs about Mixmaster dynamics, proposing explicit conjectures on asymptotic states and the relevance of the Mixmaster/Kasner map.

Findings

Formulated explicit conjectures on past asymptotic states.

Analyzed stochastic properties of the Mixmaster/Kasner map.

Compared dynamical systems approach with metric and billiard models.

Abstract

We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. Surprisingly few facts are known about the `Mixmaster' dynamics of these models, while at the same time most of the commonly held beliefs are rather vague. In this paper, we use Mixmaster facts as a base to build an infrastructure that makes it possible to sharpen the main Mixmaster beliefs. We formulate explicit conjectures concerning (i) the past asymptotic states of type IX solutions and (ii) the relevance of the Mixmaster/Kasner map for generic past asymptotic dynamics. The evidence for the conjectures is based on a study of the stochastic properties of this map in conjunction with dynamical systems techniques. We use a dynamical systems formulation, since this approach has so far been the only successful path to obtain theorems, but we also make comparisons with the `metric' and Hamiltonian `billiard' approaches.