# Bianchi VIII and IX vacuum cosmologies: Almost every solution forms particle
horizons and converges to the Mixmaster attractor

Bernhard Brehm111Free University Berlin<EMAIL_ADDRESS>
###### Abstract

Bianchi models are posited by the BKL picture to be essential building blocks
towards an understanding of generic cosmological singularities. We study the
behaviour of spatially homogeneous anisotropic vacuum spacetimes of Bianchi
type VIII and IX, as they approach the big bang singularity.

It is known since 2001 that generic Bianchi IX spacetimes converge towards the
so-called Mixmaster attractor as time goes towards the singularity. We extend
this result to the case of Bianchi VIII vacuum.

The BKL picture suggests that particle horizons should form, i.e. spatially
separate regions should causally decouple. We prove that this decoupling
indeed occurs, for Lebesgue almost every Bianchi VIII and IX vacuum spacetime.

###### Contents

  1. 1 Introduction
  2. 2 Setting, Notation and the Wainwright-Hsu equations
    1. 2.1 Spatially Homogeneous Spacetimes
    2. 2.2 The Wainwright-Hsu equations
    3. 2.3 The Wainwright-Hsu equations in polar coordinates
  3. 3 Description of the Dynamics
    1. 3.1 Lower Bianchi Types
    2. 3.2 Bianchi-Types VIII and IX for large $N$
    3. 3.3 The Bianchi IX Attractor Theorem
  4. 4 Dynamics near the Mixmaster-Attractor $\mathcal{A}$
  5. 5 Analysis near the generalized Taub-spaces $\mathcal{T}_{i}^{G}$
    1. 5.1 Informal Motivation
    2. 5.2 Analysis near $-{\mathbf{T}}_{1}$ and near the heteroclinic $-{\mathbf{T}}_{1}\to{\mathbf{T}}_{1}$
    3. 5.3 Analysis near ${\mathbf{T}}_{1}$
    4. 5.4 Analysis in the Inaccessible-cones
  6. 6 Attractor Theorems
  7. 7 Phase-Space Volume and Integral Estimates
    1. 7.1 Volume Expansion
    2. 7.2 Proofs of the main Theorems
  8. 8 Physical Properties of Solutions for Bianchi VIII and IX
  9. A Appendix
    1. A.1 Glossary of Equations and Notations
    2. A.2 Properties of the Kasner Map
    3. A.3 Transformation of Volumes on Manifolds
    4. A.4 Derivation of the Wainwright-Hsu equations

##  1 Introduction

##### Spatially homogeneous cosmological models.

The behaviour of cosmological models is governed by the Einstein field
equations, coupled with equations describing the presence of matter. Simpler
models are obtained under symmetry assumption. The class of models studied in
this work, Bianchi models, assume spatial homogeneity, i.e. “every point looks
the same”. Then, one only needs to describe the behaviour over time of any
single point, and the partial differential Einstein field equations become a
system of ordinary differential equations.

Directional isotropy assumes that “every spatial direction looks the same”.
This leads to the well-known FLRW (Friedmann-Lemaitre-Robertson-Walker)
models. These models describe an initial (“big bang”) singularity, followed by
an expansion of the universe, slowed down by ordinary and dark matter and
accelerated by a competing positive cosmological constant (“dark energy”).

We will assume spatial homogeneity, but relax the assumption of directional
isotropy. Spatial homogeneity assumes that there is a Lie-group $G$ of
spacetime isometries, which foliates the spacetime into three-dimensional
space-like hypersurfaces on which $G$ acts transitively: For every two points
${\bm{x}},{\bm{y}}$ in the same hypersurface there is a group element $f\in G$
such that $f\cdot{\bm{x}}={\bm{y}}$. The resulting ordinary differential
equations depend on the Lie-algebra of $G$, the so-called Killing fields. The
three-dimensional Lie-algebras have been classified by Luigi Bianchi in 1898,
hence the name“Bianchi-models”; for a commented translation see [Bia01,
Jan01], and for a modern treatment see [WE05, Section 1].

The two most studied classes of spatially homogeneous anisotropic cosmological
models are the Bianchi-types IX ($so(3)$) and VIII ($sl(2,{\mathbb{R}})$),
which are the focus of this work. Both of these models exhibit a big-bang like
singularity in at least one time-direction, and a universe that initially
expands from this singularity, until, in the case of Bianchi IX, it
recollapses into a time-reversed big bang (“big crunch”) The big-bang
singularity is present even in the vacuum case, where matter is absent and
only gravity self-interacts. According to conventional wisdom, “matter does
not matter” near the singularity. For this reason, we simplify our analysis by
considering only the vacuum case.

Note that the symmetry assumptions already restrict the global topology of the
space-like hypersurfaces, and that the isotropic FLRW-models are not contained
as a special case: The only homogeneous isotropic vacuum model is flat
Minkowski space.

For a detailed introduction to Bianchi models, we refer to [WE05]. A short
derivation of the governing ordinary differential Wainwright-Hsu equations
(2.2.2) is given in Section A.4, and physical interpretations of some of our
results are given in Section 8. For an excellent survey on Bianchi
cosmologies, we refer to [HU09a], and for further physical questions we refer
to [UVEWE03, HUR+09].

##### The Taub-spaces.

The dynamical behaviour of Bianchi VIII and IX spacetimes is governed by the
so-called Wainwright-Hsu equations. This dynamical system contains an
invariant set $\mathcal{T}$ of codimension two, which we call the Taub-spaces.
Solutions in this set are also called Taub-NUT spacetimes, or LRS (locally
rotationally symmetric) spacetimes. The latter name is descriptive, in the
sense that these spacetimes have additional (partial, local) isotropy. Taub
spacetimes behave, in several ways, different from general (i.e. non-Taub)
Bianchi spacetimes. For a detailed description, we refer to [Rin01].

##### The Mixmaster attractor.

The Bianchi dynamical system also contains an invariant set $\mathcal{A}$,
called the Mixmaster attractor, consisting of Bianchi Type II and I solutions.
There are good heuristic arguments that $\mathcal{A}$ really is an attractor
for time approaching the singularity, and that the dynamics on and near
$\mathcal{A}$ can be considered chaotic (sometimes also called “oscillatory”).
It has been rigorously proven only in Bianchi IX models that $\mathcal{A}$ is
actually attracting, with the exception of some Taub solutions, c.f. [Rin01].

##### Stable Foliations.

One may ask for a more precise description of how solutions get attracted to
$\mathcal{A}$. Certain heteroclinic chains in $\mathcal{A}$ are known to
attract hypersurfaces of codimension one, c.f. [LHWG11, Bég10]. Reiterer and
Trubowitz [RT10] claim related results, for a much wider class of heteroclinic
chains, but with less focus on the regularity or codimension of the attracted
sets. This general class of constructions, i.e. partial stable foliations over
specific solutions in $\mathcal{A}$, is not the focus of this work. Instead,
we describe and estimate solutions and the solution operator (i.e. flow)
directly, without explicitly focussing on the symbolic description of
$\mathcal{A}$.

##### The question of particle horizons.

One of the most salient features of relativity is _causality_ : The state of
the world at some point in spacetime is only affected by states in its past
light-cone and can only affect states in its future light-cone.

Suppose for this paragraph that we orient our spacetime $M$ such that the big
bang singularity is situated in the past. Two points ${\bm{p}},{\bm{q}}\in M$
are said to _causally decouple_ towards the singularity if their past light-
cones are disjoint, i.e. if there is no past event which causally influences
both ${\bm{p}}$ and ${\bm{q}}$. The past communication cone of ${\bm{p}}$ is
defined to be the set of points, which do not causally decouple from
${\bm{p}}$. The cosmic horizon, also called particle horizon, is the boundary
of the past communication cone. Hence, everything beyond the horizon is
causally decoupled.

Past light cone of ${\bm{p}}$: (1.1) $\displaystyle J^{-}({\bm{p}})$
$\displaystyle=\\{{\bm{q}}:\,\text{there is }\gamma:[0,1]\to
M\,\,\text{with}\,\gamma(0)={\bm{p}},\gamma(1)={\bm{q}},\,\text{time-like past
directed}\\}$ Future light cone of ${\bm{p}}$: $\displaystyle J^{+}({\bm{p}})$
$\displaystyle=\\{{\bm{q}}:\,\text{there is }\gamma:[0,1]\to
M\,\,\text{with}\,\gamma(0)={\bm{p}},\gamma(1)={\bm{q}},\,\text{time-like
future directed}\\}$ Past communication cone of ${\bm{p}}$: $\displaystyle
J^{+}(J^{-}({\bm{p}}))$ $\displaystyle=\bigcup_{{\bm{q}}\in
J^{-}({\bm{p}})}J^{+}({\bm{q}})=\\{{\bm{q}}:\,{J^{-}({\bm{q}})}\cap{J^{-}({\bm{p}})}\neq\emptyset\\}$
Past cosmic horizon of ${\bm{p}}$: $\displaystyle\partial
J^{+}(J^{-}({\bm{p}}))$
$\displaystyle=\mathrm{closure}\,J^{+}(J^{-}({\bm{p}}))\,\setminus\,\mathrm{interior}\,J^{+}(J^{-}({\bm{p}})).$

In Figure 1, we illustrate the formation of particle horizons. An example
where no particle horizons form is given by the (flat, connected) Minkowski-
space $M={\mathbb{R}}\times{\mathbb{R}}^{3}$: There is no singularity, and
$M=J^{+}(J^{-}({\bm{p}}))$ for all ${\bm{p}}$, and hence $\partial
J^{+}(J^{-}({\bm{p}}))=\emptyset$.

Apart from the question of convergence to $\mathcal{A}$, the next important
physical question in the context of Bianchi cosmologies is that of locality of
light-cones:

  1. 1.

Do nonempty particle horizons $\partial J^{+}(J^{-}({\bm{p}}))\neq\emptyset$
form towards the singularity? Does this happen if ${\bm{p}}$ is sufficiently
near to the singularity?

  2. 2.

Are the spatial hypersurfaces
$\\{t=t_{0}=\mathrm{const}\\}\cap(J^{+}(J^{-}({\bm{p}})),\partial
J^{+}(J^{-}({\bm{p}}))$, considered as three-dimensional manifolds with
boundary, homeomorphic to the three dimensional unit ball $(B_{1}(0),\partial
B_{1}(0))$?

  3. 3.

Are the past communication cones of ${\bm{p}}$ spatially bounded? Do they
shrink down to a point, as ${\bm{p}}$ and $t_{0}$ go towards the singularity?

The first question is formulated completely independently of the foliation of
the spacetime into space-like hypersurfaces. The second question depends on
the foliation, but is at least easy to clearly state. The third question is
not clearly stated here, because it requires us to choose a way of comparing
spatial extents of the communication cones at different times. The subtleties
of this are discussed in Section 8.

Since this work is concerned only with proving affirmative answers to these
questions, we will conflate them: We say that a solution forms a particle
horizon if all these questions are answered with “yes”.

(a) Two points ${\bm{p}}$, ${\bm{q}}$ that decouple towards the singularity.
Their past light-cones are disjoint.

(b) Two points ${\bm{p}}$, ${\bm{q}}$ that decouple. The point ${\bm{q}}$ lies
outside the communication cone of ${\bm{p}}$.

(c) Two points ${\bm{p}}$, ${\bm{q}}$ that do not decouple towards the
singularity. Their past light-cones have nonempty intersection (the darkest
shaded region). The point ${\bm{q}}$ lies inside the communication cone of
${\bm{p}}$.

Figure 1: Examples of decoupling and non-decoupling towards the singularity at
$t=t_{\mathrm{sing}}$, and of particle horizons.

Originally, Misner [Mis69] suggested that no particle horizons should form in
Bianchi IX. This was proposed as a possible explanation of the observed
approximate homogeneity of the universe: If the homogeneity is due to past
mixing, then different observed points in our current past light-cone must
themselves have a shared causal past. Misner later changed his mind to the
current consensus intuition that typical Bianchi VIII and IX solutions should
form particle horizons. Some more details on this are given in Section 8.
Further discussion of these questions can be found in e.g. [Wal84, Chapter 5],
[HE73, Chapter 5].

##### The BKL picture.

Spatially homogeneous spacetimes, and especially the question of particle
horizons, play an essential role in the so-called BKL picture (also often
called BKL-conjecture). The BKL picture is due to Belinskii, Khalatnikov and
Lifshitz ([BKL70]), and describes generic cosmological singularities in terms
of homogeneous spacetimes. This picture roughly claims the following:

  1. 1.

Generic cosmological singularities “are curvature-dominated”, i.e. behave like
the vacuum case. More succinctly, “matter does not matter”.

  2. 2.

Generic cosmological singularities are “oscillatory”, which means that the
directions that get stretched or compressed switch over time.

  3. 3.

Generic cosmological singularities “locally behave like” spatially homogeneous
ones, especially Bianchi IX and VIII. By this, one means that:

    1. (a)

Different regions causally decouple towards the singularity, i.e. particle
horizons form.

    2. (b)

If one restricts attention to a single communication cone, then, as time goes
towards the singularity, the spacetime can be well approximated by a
homogeneous one.

    3. (c)

Different spatial regions may have different geometry towards the singularity
(since they decouple). This kind of behaviour has been described as “foam-
like”.

Boundedness of the communication cones, i.e. formation of particle horizons,
in spatially homogeneous models is a necessary condition for the consistency
of the BKL-picture: $(3.a)$ claims that different spatial regions causally
decouple towards the big bang, and $(3.b)$ claims that such decoupled regions
behave “like they were homogeneous”; hence, homogeneous solutions should
better allow for different regions to spatially decouple.

##### Previous results.

One way of viewing the formation of particle horizons is as a race between the
expansion of the universe (shrinking towards the singularity) and the eventual
blow-up at the singularity. If the blow-up is faster than the expansion,
particle horizons form; otherwise, they don’t. In the context of the
Wainwright-Hsu equations, the question can be boiled down to: _Do solutions
converge to $\mathcal{A}$ sufficiently fast? If yes, then particle horizons
form. If not, then the questions of particle horizons may have subtle
answers._

The aforementioned solutions constructed in [LHWG11, Bég10], with initial
conditions on certain hypersurfaces of codimension one, all converge
essentially uniformly exponentially to $\mathcal{A}$, which is definitely fast
enough for particle horizons to form.

Reiterer and Trubowitz claim in [RT10] that the solutions constructed therein
also converge to $\mathcal{A}$ fast enough for this to happen. The claimed
results in [RT10] are somewhat nontrivial to parse; let us give a short
overview: They construct solutions converging to certain parts of the
Mixmaster attractor $\mathcal{A}$. These parts of the Mixmaster attractor have
full (one-dimensional) volume, and all these constructed solutions form
particle horizons. Claims about full-dimensional measure (or Hausdorff-
dimensions, etc) are not made in [RT10].

The solutions constructed in [LHWG11, Bég10] were the first known nontrivial
solutions that could be proven to form particle horizons. It is still unknown
whether there exist nontrivial counterexamples, i.e. non-Taub solutions that
fail to form particle horizons.

##### Main results.

The first main result of this work extends Ringström’s Bianchi IX attractor
theorem (Theorem 1, c.f. also [Rin01, HU09b]) to the case of Bianchi VIII
vacuum. It can be summarized in the following:

######  Theorem 2, 3, 4 and 5 (Paraphrased Attractor Theorem).

With certain exceptions, solutions in Bianchi IX and VIII vacuum converge to
the Mixmaster attractor $\mathcal{A}$.

Lower bounds on the speed of convergence are given, but are insufficient to
ensure the formation of particle horizons.

The dynamics of the exceptional solutions is described. The set of initial
conditions corresponding to the exceptional solutions is nongeneric, both in
the sense of Lebesgue (it is a set of zero Lebesgue measure) and Baire (it is
a meagre set).

Apart from the applicability to Bianchi VIII, this also extends the
Ringström’s previous result by providing lower bounds on the speed of
convergence, and provides a new proof.

The most important result of this work is the following:

######  Theorem 6 (Almost sure formation of particle horizons).

Almost every solution in Bianchi VIII and IX vacuum forms particle horizons
towards the big bang singularity. A more rigorous formulation of the theorem
is on page 6.

The question remains open of whether particle horizons form for initial
conditions that are generic in the sense of Baire222A set is called generic in
the sense of Baire, if it is _co-meagre_ , i.e. it contains a countable
intersection of open and dense sets. Then its complement is called _meagre_.
By construction, countable intersections of co-meagre sets are co-meagre and
countable unions of meagre sets are meagre. Baire’s Theorem states that co-
meagre subsets of complete metric spaces are always dense and especially
nonempty.. We strongly suspect that the answer is no, i.e. particle horizons
fail to form for a co-meagre set of initial conditions, for reasons which will
be explained in future work.

##### Structure of this work.

We will give the Wainwright-Hsu equations in Section 2, as well as some
notation and transformations that will be needed later on. The most referenced
equations are also summarized in Appendix A.1, page A.1, for easier reference.
A derivation of the Wainwright-Hsu equations from the Einstein field equations
of general relativity is given Section A.4.

An overview of the dynamical behaviour and some first proofs will be given in
Section 3. Sections 4 and 5 will describe in detail two different regimes in
the neighborhood of $\mathcal{A}$; these two descriptions are synthesized into
general attractor theorems in Section 6. The measure-theoretic results are all
contained in Section 7. Section 8 relates dynamical properties of solutions to
the Wainwright-Hsu equations to physical properties of the corresponding
spacetime.

##### Strategy.

Our analysis of the behaviour of solutions of the Wainwright-Hsu equation is
structured around two invariant objects: The Mixmaster-attractor $\mathcal{A}$
and the Taub-spaces $\mathcal{T}$. We will measure the distances from these
sets by functions $\delta({\bm{x}})\sim d({\bm{x}},\mathcal{A})$ and
$r({\bm{x}})\sim d({\bm{x}},\mathcal{T})$.

There exist standard heuristics based on normal hyperbolicity, cf. [HU09a].
According to these, solutions near $\mathcal{A}$:

  1. 1.

can be described by the so-called Kasner-map (see Section 3),

  2. 2.

converge exponentially to $\mathcal{A}$, and

  3. 3.

their associated spacetimes form particle-horizons.

These heuristics break down near the Taub-spaces $\mathcal{T}\cap\mathcal{A}$,
where two eigenvalues pass through zero.

In Section 4 we will formally prove the validity of the heuristic description
of solutions near $\mathcal{A}$, that stay bounded away from $\mathcal{T}$.
More precisely, we will show in Proposition 4.1 that for any $\epsilon_{T}>0$,
there exists $\epsilon_{d}>0$ such that all the previously mentioned
hyperbolicity heuristics apply for solutions
${\bm{x}}:[0,T]\to\\{{\bm{y}}:\,\delta({\bm{y}})<\epsilon_{d},\,r({\bm{y}})>\epsilon_{T}\\}$.

To provide a complete picture of the dynamics, we still need to control
solutions in a neighborhood of $\mathcal{T}$. This is the goal of Section 5.
It is well known, that $\mathcal{T}$ is transient, i.e. solutions may approach
$\mathcal{T}$ but cannot converge to $\mathcal{T}$; they must leave a
neighborhood of $\mathcal{T}$ again. Every component of
$\mathcal{T}\cap\mathcal{A}$ consists of two equilibria connected by a
heteroclinic orbit $-{\mathbf{T}}\to+{\mathbf{T}}$; the standard heuristics
described in Section 4 break down at $+{\mathbf{T}}$, but continue to work
near $-{\mathbf{T}}$ and near the heteroclinic orbit
$-{\mathbf{T}}\to+{\mathbf{T}}$. Thus, solutions can leave the region
controlled by normal hyperbolicity only by approaching first $-{\mathbf{T}}$
and then $+{\mathbf{T}}$.

The most important quantity in the local analysis near $+{\mathbf{T}}$ is the
quotient $\frac{\delta}{r}$ of the distances to $\mathcal{A}$ and
$\mathcal{T}$. As long as this quotient is small, we can prove exponential
decay of $\delta(t)$ (Proposition 5.3) and slow growth of $r(t)$; hence, the
quotient $\frac{\delta}{r}$ continues to decrease. The structure of this kind
of estimate is not surprising; in fact, it is trivial (by varying
$\epsilon_{d}$ and $\epsilon_{T}$) to construct continuous nonnegative
functions $\hat{\delta}$ and $\hat{r}$ with
$\mathcal{A}=\\{{\bm{x}}:\hat{\delta}({\bm{x}})=0\\}$ and
$\mathcal{T}=\\{{\bm{x}}:\hat{r}({\bm{x}})=0\\}$, such that estimates of the
above form are true near $+{\mathbf{T}}$. However, our particular choice of
functions $\delta$ and $r$ (explicitly given in Section 2.3) allows the _same_
quotient $\frac{\delta}{r}$ to be controlled near $-{\mathbf{T}}$ and near
$+{\mathbf{T}}$. We do not know of any reason to a priori expect this
fortuitous fact; it is, however, easily verified by direct calculation.

The analysis from Section 5 fits together with the analysis from Section 4:
Solutions near $\mathcal{A}$ (i.e. $\delta\ll 1$) that leave regions with
$r>\epsilon_{T}$ to enter the neighborhood of $\mathcal{T}$ with
$r\leq\epsilon_{T}$ must have, at the moment where $r=\epsilon_{T}$, a very
small quotient $\delta/r<\epsilon_{T}^{-1}\epsilon_{d}\ll 1$. This gives rise
to a local attractor Theorem 2: If
$\widetilde{\delta}=\max(\delta,\frac{\delta}{r})$ is small enough for some
initial condition, then $\widetilde{\delta}$ converges to zero. Hence, we
provide a family of forward invariant neighborhoods of
$\mathcal{A}\setminus\mathcal{T}$ attracted to $\mathcal{A}$. Using the local
attractor Theorem 2, it is rather straightforward to adapt the “global” (i.e.
away from $\mathcal{A}$) arguments from [Rin01] in order to produce the global
attractor Theorems 3 and 4.

Note that as in previous works [Rin01], a “global attractor theorem” does not
imply that _all_ solutions converge to $\mathcal{A}$, but rather it describes
the exceptions, i.e. solutions where the local attractor Theorem 2 does not
eventually hold. In Bianchi IX models, i.e. in Theorem 3, these exceptions are
exactly solutions contained in the lower-dimensional Taub-spaces $\mathcal{T}$
(originally shown in [Rin01]; we provide an alternative proof). In Bianchi
VIII models, i.e. in Theorem 4, these exceptions are either contained in the
lower-dimensional Taub-spaces $\mathcal{T}$ or must follow the very particular
asymptotics described in Theorem 4, case Except. As we show in Theorem 5, this
exceptional case Except applies at most for a set of initial conditions, which
is meagre (small Baire category) and has Lebesgue measure zero. It is
currently unknown whether this exceptional case Except is possible at all.

These genericity results (Theorem 5) rely on measure theoretic considerations
in Section 7: We provide a volume-form $\omega_{4}$, which is expanded under
the flow (given in (7.1.1), (7.1.2)). This is in itself not surprising:
Hamiltonian systems preserve their canonical volume form. Since the
Wainwright-Hsu-equations derive from the (Hamiltonian) Einstein Field
equations by an essentially monotonous time-dependent rescaling, we expect to
find a volume form $\omega_{4}$, that is essentially monotonously expanding.
Such an expanding volume form has the useful property that all forward
invariant sets must either have infinite or zero $\omega_{4}$-volume.

We prove the genericity Theorem 5 by noting that the exceptional solutions
form an invariant set; using their detailed description, we can show that its
$\omega_{4}$-volume is finite and hence zero.

The results the formation of particle horizons are also proved in Section 7.
In our language, the primary question is whether
$\int_{0}^{\infty}\delta(t){\mathrm{d}}t<\infty$. If this integral is finite,
then particle horizons form and the singularity is “local”; this behaviour is
both predicted by the BKL picture and required for its consistency.
Heuristically, time spent away from $\mathcal{T}$ helps convergence of the
integral (since $\delta$ decays uniformly exponentially in these regions),
while time spent near $\mathcal{T}$ gives large contributions to the integral.
Looking at our analysis near $+{\mathbf{T}}$ (Proposition 5.3), we get a
contribution to the integral of order $\frac{\delta}{r^{2}}$. The local
attractor Theorem 2 is insufficient to decide the question of locality, since
it can only control the quotient $\frac{\delta}{r}$ (and hence the integral
$\int\delta^{2}{\mathrm{d}}t<\infty$).

However, we can show by elementary calculation that the set
$\\{{\bm{x}}:\,\delta>r^{4}\\}$ has finite $\omega_{4}$-measure. Using some
uniformity estimates on the volume expansion, we can infer that the (naturally
invariant) set Bad of initial conditions that fail to have $\delta<r^{4}$ for
all sufficiently large times, has finite and hence vanishing
$\omega_{4}$-volume. Therefore, for Lebesgue almost every initial condition,
$\delta<r^{4}$ holds eventually, allowing us to bound the contribution of the
entire stay near $+{\mathbf{T}}$ by $\frac{\delta}{r^{2}}<\sqrt{\delta}$,
which decays exponentially in the “number of episodes near Taub-points” (also
called Kasner-eras). This gives rise to Theorem 6: Lebesgue almost every
initial condition in Bianchi VIII and IX forms particle horizons towards the
big bang singularity.

A finer look at the integral even allows us to bound certain
$L^{p}_{\textrm{loc}}(\omega_{4})$ integrals in Theorem 7.

##### Acknowledgements

This work has been partially supported by the Sonderforschungsbereich “Raum,
Zeit, Materie”. This is a preliminary version of a dissertation thesis. For
helpful comments and discussions, I would like to thank, in lexicographic
order: Lars Andersson, Bernold Fiedler, Hanne Hardering, Julliette Hell,
Stefan Liebscher, Alan Rendall, Hans Ringström, and Claes Uggla.

Comments, especially those pointing out major or minor errors, are
particularly welcome.

##  2 Setting, Notation and the Wainwright-Hsu equations

The subject of this work, i.e., the behaviour of homogeneous anisotropic
vacuum space-times with Bianchi Class A homogeneity under the Einstein field
equations of general relativity, can be described by a system of ordinary
differential equations, called the Wainwright-Hsu equations (2.2.4).

In Section 2.1, we will introduce the Wainwright-Hsu ordinary differential
equations and various auxiliary quantities and definitions, and provide a
rough summary of their dynamics. Then we transform the Wainwright-Hsu
equations into polar coordinates in Section 2.3, which are essential for the
analysis in Section 5.

There are multiple equivalent formulations of the Wainwright-Hsu equations in
use by different authors, which differ in sign and scaling conventions, most
importantly the direction of time. This work uses reversed time, such that the
big bang singularity is at $t=+\infty$. The relation of the Wainwright-Hsu
equations to the Einstein equations of general relativity will be relegated to
Section A.4. The relation between properties of solutions to the Wainwright-
Hsu equations and physical properties of the corresponding spacetimes is
discussed in Section A.4.

##### General Notations.

In this work, we will often use the notation ${\bm{x}}=(x_{1},\ldots,x_{n})$
in order to emphasize that a variable ${\bm{x}}$ refers to a point and not to
a scalar quantity. If we consider a curve ${\bm{x}}(t)$ into a space where
different coordinates have names, e.g.
${\bm{x}}:{\mathbb{R}}\to{\mathbb{R}}^{5}=\\{(\Sigma_{+},\Sigma_{-},N_{1},N_{2},N_{3})\\}$,
then we will in an abuse of notation write $N_{1}(t)=N_{1}({\bm{x}}(t))$ in
order to refer to the $N_{1}$-coordinate of ${\bm{x}}(t)$.

We will use $\pm$ to refer to either $+1$ or $-1$, and different occurrences
of $\pm$ are always unrelated, such that e.g.
$(\pm,\pm,\pm)\in\\{(+,+,+),(-,+,+),(+,-,+),(-,-,+),\ldots\\}$. We will use
$*$ to refer to either $+1$,$-1$ or $0$, also such that different occurrences
of $*$ are unrelated.

###  2.1 Spatially Homogeneous Spacetimes

We study the behaviour of homogeneous spacetimes, also called Bianchi-models.
These are Lorentz four-manifolds, foliated by space-like hypersurfaces on
which a group of isometries acts transitively, subject to the vacuum Einstein
Field equations. That is, we assume that we have a frame of four linearly
independent vectorfields $e_{0}=\partial_{t},e_{1},e_{2},e_{3}$, where
$e_{1},e_{2},e_{3}$ are Killing fields, with dual co-frame
${\mathrm{d}}t,\omega_{1},\omega_{2},\omega_{3}$, such that the metric has the
form

$g=g_{00}(t){\mathrm{d}}t\otimes{\mathrm{d}}t+g_{11}(t)\omega_{1}\otimes\omega_{1}+g_{22}(t)\omega_{2}\otimes\omega_{2}+g_{33}(t)\omega_{3}\otimes\omega_{3},$

and the commutators (i.e. the Lie-algebra of the spatial homogeneity) has the
form

$[e_{i},e_{j}]=\sum_{k}\gamma_{ij}^{k}e_{k}\qquad\gamma_{ij}^{k}={{\hat{n}}}_{k}\epsilon_{ijk},$

where $\epsilon_{ijk}$ is the usual Levi-Civita symbol ($\epsilon_{ijk}=+1$ if
$(ijk)\in\\{(123),(231),(312)\\}$, $\epsilon_{ijk}=-1$ if
$(ijk)\in\\{(132),(321),(213)\\}$ and $\epsilon_{ijk}=0$ otherwise). The signs
${{\hat{n}}}_{i}\in\\{+1,-1,0\\}$ determine the Bianchi Type of the
cosmological model, according to Table 1.

The metric is described by the seven Hubble-normalized variables
$H,\widetilde{N}_{i},\Sigma_{i}$, with $i\in\\{1,2,3\\}$, according to

$g_{00}=-\frac{1}{4}H^{-2}\qquad
g_{ii}=\frac{1}{48}\frac{H^{-2}}{\widetilde{N}_{j}\widetilde{N}_{k}},$ (2.1.1)

where $(i,j,k)$ is always assumed to be a permutation of $\\{1,2,3\\}$, and
subject to the linear and sign constraints

$\Sigma_{1}+\Sigma_{2}+\Sigma_{3}=0,\qquad
H<0,\qquad\widetilde{N}_{i}>0\quad\text{for all }i\in\\{1,2,3\\}.$ (2.1.2)

The variable $H$ corresponds to the Hubble scalar, i.e. the expansion speed of
the cosmological model, i.e. the mean curvature of the surfaces
$\\{t=\textrm{const}\\}$ of spatial homogeneity. The “shears” $\Sigma_{i}$
correspond to the trace-free Hubble-normalized principal curvatures (hence,
the linear “trace-free” constraint). The condition $H<0$ corresponds to our
choice of the direction of time: We choose to orient time such that the
universe is shrinking, i.e. the singularity (big bang) lies in the future;
this unphysical choice of time-direction is just for convenience of notation.

The vacuum Einstein Field equations state that the space-time is Ricci-flat.
If we express the normalized trace-free principal curvatures $\Sigma_{i}$ as
time-derivatives of the metric variables $\widetilde{N}_{i}$, then the the
Einstein Field equations become the Wainwright-Hsu equations (2.1.3), which
are a system of seven ordinary differential equations, subject to one linear
constraint equation ($\Sigma_{1}+\Sigma_{2}+\Sigma_{3}=0$) and one algebraic
equation, called the Gauss-constraint $G=1$ (2.1.2)):

$\displaystyle H^{\prime}$ $\displaystyle=\frac{1}{2}(1+2\Sigma^{2})H$ (2.1.3)
$\displaystyle\Sigma_{i}^{\prime}$
$\displaystyle=(1-\Sigma^{2})\Sigma_{i}+\frac{1}{2}S_{i}$
$\displaystyle\widetilde{N}_{i}^{\prime}$
$\displaystyle=-(\Sigma^{2}+\Sigma_{i})\widetilde{N}_{i}$ $\displaystyle 1$
$\displaystyle\overset{!}{=}\Sigma^{2}+N^{2}:=G,$

where we used the shorthands

$\displaystyle N_{i}$ $\displaystyle:=\hat{n}_{i}\widetilde{N}_{i}$
$\displaystyle\Sigma^{2}$
$\displaystyle:=\frac{1}{6}(\Sigma_{1}^{2}+\Sigma_{2}^{2}+\Sigma_{3}^{2})$
$\displaystyle N^{2}$
$\displaystyle:=N_{1}^{2}+N_{2}^{2}+N_{3}^{2}-2(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{1})$
$\displaystyle S_{i}$
$\displaystyle:=4\left(N_{i}(2N_{i}-N_{j}-N_{k})-(N_{j}-N_{k})^{2}\right).$

$\begin{array}[]{cclcrclc}{{\hat{n}}}_{1}&{{\hat{n}}}_{2}&{{\hat{n}}}_{3}&\text{Bianchi
Type}&\qquad{{\hat{n}}}_{1}&{{\hat{n}}}_{2}&{{\hat{n}}}_{3}&\text{Bianchi
Type}\\\ +&+&+&\text{{IX}}&\qquad+&-&+&\text{{VIII}}\\\
+&+&0&\text{$\textsc{VII}_{0}$}&+&-&0&\text{$\textsc{VI}_{0}$}\\\
+&0&0&\text{{II}}&0&0&0&\text{{I}}.\end{array}$

Table 1: Bianchi Types of Class A, depending on ${{\hat{n}}}_{i}$ (up to
permutation and simultaneous sign-reversal)

###  2.2 The Wainwright-Hsu equations

The equation for $H$ is decoupled from the remaining equations. Thus, we can
drop the equation for $H$, solve the remaining equations, and afterwards
integrate to obtain $H$. Likewise, we can stick with the equations for $N_{i}$
instead of $\widetilde{N}_{i}$, such that $\hat{n}_{i}=\mathrm{sign}\,N_{i}$;
for Bianchi-types VIII and IX this already determines the metric, and for the
lower Bianchi types we can again integrate afterwards. This yields a standard
form of the Wainwright-Hsu equations from (2.1.3), as used in [HU09a],
[HU09b], up to constant factors. The most useful equations are also summarized
in Section A.1.

It is useful to solve for the linear constraint
$\Sigma_{1}+\Sigma_{2}+\Sigma_{3}=0$, introducing
${\bm{\Sigma}}=(\Sigma_{+},\Sigma_{-})$ by

$\begin{gathered}{\mathbf{T}}_{1}=(-1,0)\qquad{\mathbf{T}}_{2}=\left(\frac{1}{2},-\frac{1}{2}\sqrt{3}\right)\qquad{\mathbf{T}}_{3}=\left(\frac{1}{2},\frac{1}{2}\sqrt{3}\right)\\\
\Sigma_{i}=2\langle{\mathbf{T}}_{i},{\bm{\Sigma}}\rangle\qquad\Sigma_{+}=-\frac{1}{2}\Sigma_{1}\qquad\Sigma_{-}=\frac{1}{2\sqrt{3}}(\Sigma_{3}-\Sigma_{2}),\end{gathered}$
(2.2.1)

which turns the vacuum Wainwright-Hsu differential equations into a system of
five ordinary differential equations on
${\mathbb{R}}^{5}=\\{(\Sigma_{+},\Sigma_{-},N_{1},N_{2},N_{3})\\}=\\{{\bm{\Sigma}},{\bm{N}}\\}$,
with one algebraic constraint equation (2.2.3). The three points
${\mathbf{T}}_{1},{\mathbf{T}}_{2},{\mathbf{T}}_{3}$ are called Taub-points.
We will, in an abuse of notation, consider the Taub-points both as points in
${\mathbb{R}}^{2}$, and as points in ${\mathbb{R}}^{5}$ (where all three
$N_{i}$ vanish). The Wainwright-Hsu equations are then given by the
differential equations

$\displaystyle N_{i}^{\prime}$
$\displaystyle=-(\Sigma^{2}+2\langle{\mathbf{T}}_{i},{\bm{\Sigma}}\rangle)N_{i}$
(2.2.2a)
$\displaystyle=-\left(\left|{\bm{\Sigma}}+{\mathbf{T}}_{i}\right|^{2}-1\right)N_{i}$
(2.2.2b) $\displaystyle{\bm{\Sigma}}^{\prime}$
$\displaystyle=N^{2}{\bm{\Sigma}}+2\left(N_{1}^{2}{\mathbf{T}}_{1}+N_{2}^{2}{\mathbf{T}}_{2}+N_{3}^{2}{\mathbf{T}}_{3}+N_{1}N_{2}{\mathbf{T}}_{3}+N_{2}N_{3}{\mathbf{T}}_{1}+N_{3}N_{1}{\mathbf{T}}_{2}\right)$
(2.2.2c)
$\displaystyle=N^{2}{\bm{\Sigma}}+2\left(\begin{array}[]{ccc}{\mathbf{T}}_{1}&{\mathbf{T}}_{3}&{\mathbf{T}}_{2}\\\
&{\mathbf{T}}_{2}&{\mathbf{T}}_{1}\\\
&&{\mathbf{T}}_{3}\end{array}\right)[{\bm{N}},{\bm{N}}],$ (2.2.2g)

and the Gauss constraint equation

$1\overset{!}{=}\Sigma^{2}+N^{2}=:G({\bm{x}}),$ (2.2.3)

where we used the shorthands

$\Sigma^{2}=\Sigma_{+}^{2}+\Sigma_{-}^{2},\qquad
N^{2}=N_{1}^{2}+N_{2}^{2}+N_{3}^{2}-2(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{1}).$

We can unpack these equations with unambiguous notation into

$\displaystyle N_{1}^{\prime}$ $\displaystyle=-(\Sigma^{2}-2\Sigma_{+})N_{1}$
(2.2.4a) $\displaystyle N_{2}^{\prime}$
$\displaystyle=-(\Sigma^{2}+\Sigma_{+}-\sqrt{3}\Sigma_{-})N_{2}$ (2.2.4b)
$\displaystyle N_{3}^{\prime}$
$\displaystyle=-(\Sigma^{2}+\Sigma_{+}+\sqrt{3}\Sigma_{-})N_{3}$ (2.2.4c)
$\displaystyle\Sigma_{+}^{\prime}$
$\displaystyle=N^{2}\Sigma_{+}-2N_{1}^{2}+N_{2}^{2}+N_{3}^{2}+N_{1}N_{2}-2N_{2}N_{3}+N_{1}N_{3}$
(2.2.4d) $\displaystyle\Sigma_{-}^{\prime}$
$\displaystyle=N^{2}\Sigma_{-}+\sqrt{3}\left(-N_{2}^{2}+N_{3}^{2}+N_{1}N_{2}-N_{1}N_{3}\right),$
(2.2.4e)

which is, up to constant factors, the form of the Wainwright-Hsu equations
used in [Rin01], [LHWG11] and [Bég10].

It is occasionally useful to fully tensorize the Wainwright-Hsu equations,
yielding the form

$\displaystyle{\bm{N}}^{\prime}$
$\displaystyle=-\langle{\bm{\Sigma}},{\bm{\Sigma}}\rangle{\bm{N}}-{\bm{D}}[{\bm{\Sigma}},{\bm{N}}]$
(2.2.5) $\displaystyle{\bm{\Sigma}}^{\prime}$
$\displaystyle=Q[{\bm{N}},{\bm{N}}]{\bm{\Sigma}}+{\bm{T}}[{\bm{N}},{\bm{N}}]$
$\displaystyle G({\bm{\Sigma}},{\bm{N}})$
$\displaystyle=\langle{\bm{\Sigma}},{\bm{\Sigma}}\rangle+Q[{\bm{N}},{\bm{N}}]\overset{!}{=}1,$

where $Q:{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}\to{\mathbb{R}}$ and
${\bm{T}}:{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}\to{\mathbb{R}}^{2}$ and
${\bm{D}}:{\mathbb{R}}^{2}\times{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$. We write
$Q$ as a $3\times 3$-matrix with entries in ${\mathbb{R}}$ such that
$Q[{\bm{N}},{\bm{M}}]={\bm{N}}^{T}Q{\bm{N}}=N^{2}$; we write ${\bm{T}}$ as a
similar $3\times 3$-matrix with entries in ${\mathbb{R}}^{2}$. We write
${\bm{D}}$ as a $3\times 3$-matrix with entries in ${\mathbb{R}}^{2}$ such
that ${\bm{D}}[{\bm{\Sigma}},{\bm{N}}]=({\bm{D}}{\bm{N}})\cdot{\bm{\Sigma}}$,
where ${\bm{D}}{\bm{N}}$ is the usual matrix product (with entries in
${\mathbb{R}}^{2}$) and the dot-product is evaluated component wise. Then the
tensors $Q,{\bm{T}},{\bm{D}}$ can be written as

$\displaystyle Q=\left(\begin{array}[]{ccc}1&-2&-2\\\ &1&-2\\\
&&1\end{array}\right)\qquad{\bm{T}}=\left(\begin{array}[]{ccc}2{\mathbf{T}}_{1}&2{\mathbf{T}}_{3}&2{\mathbf{T}}_{2}\\\
&2{\mathbf{T}}_{2}&2{\mathbf{T}}_{1}\\\
&&2{\mathbf{T}}_{3}\end{array}\right)\qquad{\bm{D}}=\left(\begin{array}[]{ccc}2{\mathbf{T}}_{1}&&\\\
&2{\mathbf{T}}_{2}&\\\ &&2{\mathbf{T}}_{3}\end{array}\right).$ (2.2.6)

##### Permutation Equivariance.

The equations (2.1.3) are equivariant under permutations
$\sigma:\\{1,2,3\\}\to\\{1,2,3\\}$ of the three indices. This permutation
invariance also applies to (2.2.2) as

$({\bm{\Sigma}},N_{1},N_{2},N_{3})\to(A_{\sigma}{\bm{\Sigma}},N_{\sigma(1)},N_{\sigma(2)},N_{\sigma(3)}),$

where $A_{\sigma}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ is the linear isometry
with $A_{\sigma}^{T}{\mathbf{T}}_{i}={\mathbf{T}}_{\sigma(i)}$. The equations
are also equivariant under
$({\bm{\Sigma}},{\bm{N}})\to({\bm{\Sigma}},-{\bm{N}})$, as can be seen
directly from (2.2.5).

##### Invariance of the Constraint.

The signs of the $N_{i}$ are preserved under the flow, because
$N_{i}^{\prime}$ is a multiple of $N_{i}$ and therefore $N_{i}=0$ implies
$N_{i}^{\prime}=0$. The quantity $G$ is preserved under the flow. This can
best be seen from (2.2.5):

$\displaystyle{\mathrm{D}}_{t}G$
$\displaystyle=2\langle{\bm{\Sigma}},{\bm{\Sigma}}^{\prime}\rangle+Q[{\bm{N}},{\bm{N}}^{\prime}]+Q[{\bm{N}}^{\prime},{\bm{N}}]$
$\displaystyle=2Q[{\bm{N}},{\bm{N}}]\langle{\bm{\Sigma}},{\bm{\Sigma}}\rangle+2{\bm{\Sigma}}\cdot{\bm{T}}[{\bm{N}},{\bm{N}}]-2\langle{\bm{\Sigma}},{\bm{\Sigma}}\rangle
Q[{\bm{N}},{\bm{N}}]$
$\displaystyle\qquad-Q[{\bm{D}}[{\bm{\Sigma}},{\bm{N}}],{\bm{N}}]-Q[{\bm{N}},{\bm{D}}[{\bm{\Sigma}},{\bm{N}}]]$
$\displaystyle=\Sigma\cdot{\bm{N}}^{T}\left[2{\bm{T}}-{\bm{D}}^{T}Q-Q{\bm{D}}\right]{\bm{N}}.$

Using ${\mathbf{T}}_{1}+{\mathbf{T}}_{2}+{\mathbf{T}}_{3}=0$ and (2.2.6), it
is a simple matter of matrix multiplication to verify that
$2{\bm{T}}-{\bm{D}}^{T}Q-Q{\bm{D}}=0$ and hence ${\mathrm{D}}_{t}G=0$.
Therefore, sets of the form $\\{{\bm{x}}\in{\mathbb{R}}^{5}:G({\bm{x}})=c\\}$
are invariant for any $c\in{\mathbb{R}}$ and especially for the physical
$c=1$.

The set $\mathcal{M}=\\{{\bm{x}}\in{\mathbb{R}}^{5}:G({\bm{x}})=1\\}$ is a
smooth embedded submanifold; this is apparent from the implicit function
theorem, since (if ${\bm{x}}\neq 0$)

$\begin{multlined}\frac{1}{2}{\mathrm{d}}G=\Sigma_{+}{\mathrm{d}}\Sigma_{+}+\Sigma_{-}{\mathrm{d}}\Sigma_{-}\\\
+(N_{1}-N_{2}-N_{3}){\mathrm{d}}N_{1}+(N_{2}-N_{3}-N_{1}){\mathrm{d}}N_{2}+(N_{3}-N_{1}-N_{2}){\mathrm{d}}N_{1}\neq
0.\end{multlined}\frac{1}{2}{\mathrm{d}}G=\Sigma_{+}{\mathrm{d}}\Sigma_{+}+\Sigma_{-}{\mathrm{d}}\Sigma_{-}\\\
+(N_{1}-N_{2}-N_{3}){\mathrm{d}}N_{1}+(N_{2}-N_{3}-N_{1}){\mathrm{d}}N_{2}+(N_{3}-N_{1}-N_{2}){\mathrm{d}}N_{1}\neq
0.$

##### Named invariant sets.

There are several recurring important sets, which require names and are listed
in Table 2.

$\displaystyle\mathcal{M}$
$\displaystyle=\\{{\bm{x}}\in{\mathbb{R}}^{5}:\;G({\bm{x}})=1\\}$ the
physically relevant Phase-space $\displaystyle\mathcal{M}_{{{\hat{n}}}}$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:\;\mathrm{sign}\,N_{i}={{\hat{n}}}_{i}\\}$
a specific octant of the Phase-space $\displaystyle\mathcal{K}$
$\displaystyle=\mathcal{M}_{000}=\\{{\bm{x}}\in\mathcal{M}:\;{\bm{N}}=0\\}$
the Kasner circle $\displaystyle\mathcal{A}$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:\;\text{at most one }N_{i}\neq 0\\}$
the Mixmaster attractor $\displaystyle\mathcal{A}_{{{\hat{n}}}}$
$\displaystyle=\overline{\mathcal{M}_{{{\hat{n}}}}}\cap\mathcal{A}$ a specific
octant of $\mathcal{A}$ $\displaystyle\mathcal{T}_{i}$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:N_{j}=N_{k},\,\langle{\mathbf{T}}_{j},{\bm{\Sigma}}\rangle=\langle{\mathbf{T}}_{k},{\bm{\Sigma}}\rangle\\}$
a Taub-space $\displaystyle\mathcal{T}$
$\displaystyle=\mathcal{T}_{1}\cup\mathcal{T}_{2}\cup\mathcal{T}_{3}$ all
three Taub-spaces $\displaystyle\mathcal{TL}_{i}$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:N_{j}=N_{k},\,N_{i}=0,\,{\bm{\Sigma}}={\mathbf{T}}_{i}\\}\subseteq\mathcal{T}_{i}$
a Taub-line $\displaystyle\mathcal{T}^{G}_{i}$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:|N_{j}|=|N_{k}|,\,\langle{\mathbf{T}}_{j},{\bm{\Sigma}}\rangle=\langle{\mathbf{T}}_{k},{\bm{\Sigma}}\rangle\\}$
a generalized Taub-space; only invariant if
$\mathrm{sign}\,N_{j}=\mathrm{sign}\,N_{k}$

Table 2: Named subsets. Here $(i,j,k)$ stands for a permutation of
$\\{1,2,3\\}$ and ${{\hat{n}}}\in\\{+,0,-\\}^{3}$. All of these sets, except
for $\mathcal{T}^{G}_{i}$, are invariant.

The set $\mathcal{M}$ is invariant because $G$ is a constant of motion. The
Taub-space $\mathcal{T}_{i}$ is invariant because of the equivariance under
exchange of the two other indices $j$ and $k$. The invariance of the Taub-
lines $\mathcal{TL}_{i}$ can be seen by considering (2.2.4) for $i=1$ and
applying the permutation invariance for $\mathcal{TL}_{2}$ and
$\mathcal{TL}_{3}$. The generalized Taub-spaces $\mathcal{T}^{G}_{i}$ are not
invariant if ${{\hat{n}}}_{j}\neq{{\hat{n}}}_{k}$. The other sets are
invariant because the signs ${{\hat{n}}}_{i}=\mathrm{sign}\,N_{i}$ are fixed.

Recall that the signs of the $N_{i}$ correspond to the Bianchi Type of the
Lie-algebra associated to the homogeneity of the cosmological model and are
given in Table 1 (up to index permutations).

##### Auxiliary Quantities.

The following quantities turn out to be useful later on (where $(i,j,k)$ is a
permutation of $(1,2,3)$):

$\displaystyle\delta_{i}$ $\displaystyle=2\sqrt{|N_{j}N_{k}|}$ (2.2.7a)
$\displaystyle r_{i}$
$\displaystyle=\sqrt{(|N_{j}|-|N_{k}|)^{2}+\frac{1}{3}\langle{\mathbf{T}}_{j}-{\mathbf{T}}_{k},{\bm{\Sigma}}\rangle^{2}}$
(2.2.7b) $\displaystyle\psi_{i}$ such that: $\displaystyle r_{1}\cos\psi_{1}$
$\displaystyle=\frac{1}{\sqrt{3}}\langle{\mathbf{T}}_{3}-{\mathbf{T}}_{2},{\bm{\Sigma}}\rangle=\Sigma_{-}$
$\displaystyle r_{1}\sin\psi_{1}$ $\displaystyle=|N_{2}|-|N_{3}|$ (2.2.7c)
$\displaystyle r_{2}\cos\psi_{2}$
$\displaystyle=\frac{1}{\sqrt{3}}\langle{\mathbf{T}}_{1}-{\mathbf{T}}_{3},{\bm{\Sigma}}\rangle=-\frac{\sqrt{3}}{2}\Sigma_{+}-\frac{1}{2}\Sigma_{-}$
$\displaystyle r_{2}\sin\psi_{2}$ $\displaystyle=|N_{3}|-|N_{1}|$ (2.2.7d)
$\displaystyle r_{3}\cos\psi_{3}$
$\displaystyle=\frac{1}{\sqrt{3}}\langle{\mathbf{T}}_{2}-{\mathbf{T}}_{1},{\bm{\Sigma}}\rangle=\frac{\sqrt{3}}{2}\Sigma_{+}-\frac{1}{2}\Sigma_{-}$
$\displaystyle r_{3}\sin\psi_{3}$ $\displaystyle=|N_{1}|-|N_{2}|.$ (2.2.7e)

The auxiliary products $\delta_{i}^{2}$ can be used to measure the distance
from the Mixmaster attractor
$\mathcal{A}=\\{{\bm{x}}:\,\max_{i}\delta_{i}({\bm{x}})=0\\}$. The $r_{i}$ and
can be used to measure the distance from the generalized Taub-space
$\mathcal{T}_{i}^{G}=\\{{\bm{x}}:\,r_{i}({\bm{x}})=0\\}$, and the
$(r_{i},\psi_{i})$-pairs form polar coordinates around the generalized Taub-
spaces.

The products $\delta_{i}$ obey an especially geometric differential equation,
similar to (2.2.2b):

$\delta_{i}^{\prime}=-\left(\left|{\bm{\Sigma}}-\frac{{\mathbf{T}}_{i}}{2}\right|^{2}-\frac{1}{4}\right)\delta_{i}.$
(2.2.8)

###  2.3 The Wainwright-Hsu equations in polar coordinates

Near the generalized Taub-spaces $\mathcal{T}^{G}_{i}$, it is possible to use
polar coordinates (2.2.7). Without loss of generality we will only transform
(2.2.2) into these coordinates around the Taub-space $\mathcal{T}^{G}_{1}$
(the other ones can be obtained by permuting the indices and rotating or
reflecting ${\bm{\Sigma}}$).

The use of polar coordinates near the Taub-spaces $\mathcal{T}_{i}$ for
Bianchi IX, i.e. $\mathcal{M}_{+++}$, is by no means novel (c.f. e.g. [Rin01],
[HU09a]). However, to the best of our knowledge, polar coordinates around the
generalized Taub-spaces $\mathcal{T}^{G}_{i}$ have not been used previously in
the case where the generalized Taub-space fails to be invariant.

We only use use polar coordinates on
$\mathcal{M}=\\{{\bm{x}}:\,\,G({\bm{x}})=1\\}$.

##### Polar Coordinates around the invariant Taub-spaces.

Consider the case $\mathcal{M}_{*++}$, where $N_{2},N_{3}>0$ are positive and
we are interested in a neighborhood of
$\mathcal{T}_{1}=\\{{\bm{x}}:\,\Sigma_{-}=0,\,N_{2}-N_{3}=0\\}$. The sign of
$N_{1}$ does not significantly matter.

We use the additional shorthands

$N_{-}=N_{2}-N_{3}\qquad N_{+}=N_{2}+N_{3},$

such that (with (2.2.7)):

$\displaystyle r_{1}\geq 0$ $\displaystyle:$ $\displaystyle r_{1}^{2}$
$\displaystyle=\Sigma_{-}^{2}+N_{-}^{2}$ $\displaystyle\psi$ $\displaystyle:$
$\displaystyle N_{-}$ $\displaystyle=r_{1}\sin\psi$ $\displaystyle\Sigma_{-}$
$\displaystyle=r_{1}\cos\psi$ $\displaystyle N_{+}^{2}$
$\displaystyle=N_{-}^{2}+\delta_{1}^{2}$ $\displaystyle N^{2}$
$\displaystyle=N_{-}^{2}+N_{1}(N_{1}-2N_{+}).$

This gives us the differential equations (using $\Sigma^{2}+N^{2}=1$):

$\displaystyle N_{-}^{\prime}$
$\displaystyle=(N^{2}-1-\Sigma_{+})N_{-}+\sqrt{3}\Sigma_{-}N_{+}$
$\displaystyle N_{+}^{\prime}$
$\displaystyle=(N^{2}-1-\Sigma_{+})N_{+}+\sqrt{3}\Sigma_{-}N_{-}$
$\displaystyle\Sigma_{-}^{\prime}$
$\displaystyle=N^{2}\Sigma_{-}-\sqrt{3}N_{-}\left(N_{+}-N_{1}\right),$

allowing us to further compute

$\displaystyle\frac{r_{1}^{\prime}}{r_{1}}$
$\displaystyle=\frac{\Sigma_{-}\Sigma_{-}^{\prime}+N_{-}N_{-}^{\prime}}{r_{1}^{2}}=N^{2}-(\Sigma_{+}+1)\frac{N_{-}^{2}}{r_{1}^{2}}+\sqrt{3}N_{1}\frac{\Sigma_{-}N_{-}}{r_{1}^{2}}$
(2.3.1a) $\displaystyle\psi^{\prime}$
$\displaystyle=\frac{\Sigma_{-}N_{-}^{\prime}-N_{-}\Sigma_{-}^{\prime}}{r_{1}^{2}}=\sqrt{3}N_{+}-(\Sigma_{+}+1)\frac{N_{-}\Sigma_{-}}{r_{1}^{2}}-\sqrt{3}N_{1}\frac{N_{-}^{2}}{r_{1}^{2}}$
(2.3.1b) $\displaystyle\frac{\delta_{1}^{\prime}}{\delta_{1}}$
$\displaystyle=N^{2}-(\Sigma_{+}+1)$ (2.3.1c)
$\displaystyle\partial_{t}\log\frac{\delta_{1}}{r_{1}}$
$\displaystyle=-(\Sigma_{+}+1)\frac{\Sigma_{-}^{2}}{r_{1}^{2}}-\sqrt{3}N_{1}\frac{\Sigma_{-}N_{-}}{r_{1}^{2}}.$
(2.3.1d)

Near ${\mathbf{T}}_{1}$, i.e. for $\Sigma_{+}\approx-1$, we can use
$1+\Sigma_{+}=\frac{\Sigma_{-}^{2}+N^{2}}{1-\Sigma_{+}}$ in order to rearrange
some terms in (2.3.1):

$\displaystyle\frac{r_{1}^{\prime}}{r_{1}}$
$\displaystyle=r_{1}^{2}\sin^{2}\psi\frac{-\Sigma_{+}}{1-\Sigma_{+}}+N_{1}\,h_{r}$
(2.3.2a) $\displaystyle\frac{\delta_{1}^{\prime}}{\delta_{1}}$
$\displaystyle=\frac{-1}{1-\Sigma_{+}}r_{1}^{2}\cos^{2}\psi+\frac{-\Sigma_{+}}{1-\Sigma_{+}}r_{1}^{2}\sin^{2}\psi+N_{1}\,h_{\delta}$
(2.3.2b) $\displaystyle\partial_{t}\log\frac{\delta_{1}}{r_{1}}$
$\displaystyle=\frac{-1}{1-\Sigma_{+}}r_{1}^{2}\cos^{2}\psi+N_{1}(h_{\delta}-h_{r})$
(2.3.2c) $\displaystyle\psi^{\prime}$
$\displaystyle=\sqrt{3}r_{1}\sqrt{\sin^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}-\frac{r_{1}^{2}}{1-\Sigma_{+}}\sin\psi\cos\psi+N_{1}\sin\psi\,h_{\psi},$
(2.3.2d)

where

$\displaystyle h_{r}$
$\displaystyle=+\sqrt{3}\frac{\Sigma_{-}N_{-}}{r_{1}^{2}}+(N_{1}-2N_{+})\left(1+\frac{N_{-}^{2}}{(1-\Sigma_{+})r_{1}^{2}}\right)$
(2.3.3a) $\displaystyle h_{\delta}$
$\displaystyle=(N_{1}-2N_{+})\frac{-\Sigma_{+}}{1-\Sigma_{+}}$ (2.3.3b)
$\displaystyle h_{\psi}$
$\displaystyle=-\sqrt{3}\sin\psi-\cos\psi\frac{N_{1}-2N_{+}}{1-\Sigma_{+}}.$
(2.3.3c)

Let us point out that $|h_{r}|,|h_{\delta}|,|h_{\psi}|\leq 5$ if
$|N_{1}|,|N_{+}|,|N_{-}|\leq\frac{1}{2}$ and $\Sigma_{+}<0$.

##### Polar Coordinates around the non-invariant generalized Taub-spaces.

Without loss of generality, assume that we are in $\mathcal{M}_{*+-}$, i.e.
$N_{2}>0>N_{3}$, and that we are interested in a neighborhood of
$\mathcal{T}^{G}_{1}=\\{{\bm{x}}:\,\Sigma_{-}=0,\,|N_{2}|-|N_{3}|=0\\}$.

The assumption $N_{2}>0>N_{3}$ is incompatible with $N_{3}=N_{2}$; hence,
$\mathcal{M}_{*+-}\cap\mathcal{T}_{1}=\emptyset$, which is why we have to work
with $\mathcal{T}^{G}_{1}$. Since $\mathcal{T}^{G}_{1}$ is not invariant, we
expect the corresponding equations for $r_{1}^{\prime}$ and $\psi^{\prime}$ to
become singular near $r_{1}=0$.

There is a possible symmetry-based motivation for expecting usable polar
coordinates around $\mathcal{T}^{G}_{1}$ that will be given at the end of this
section. Regardless of this motivation, the usability of such polar
coordinates is proven by our use of them.

We will proceed analogous to the case of $\mathcal{T}_{1}$, using the same
names for quantities which fulfill the same function in this work, such that
the definitions of e.g. $N_{+},N_{-}$ will depend on the signs
${{\hat{n}}}_{2},{{\hat{n}}}_{3}$. Hence, we introduce shorthands

$N_{-}=|N_{2}|-|N_{3}|=N_{2}+N_{3}\qquad N_{+}=|N_{2}|+|N_{3}|=N_{2}-N_{3},$

such that (with (2.2.7)):

$\displaystyle r_{1}\geq 0$ $\displaystyle:$ $\displaystyle r_{1}^{2}$
$\displaystyle=\Sigma_{-}^{2}+N_{-}^{2}$ $\displaystyle\psi$ $\displaystyle:$
$\displaystyle N_{-}$ $\displaystyle=r_{1}\sin\psi$ $\displaystyle\Sigma_{-}$
$\displaystyle=r_{1}\cos\psi$ $\displaystyle N_{+}^{2}$
$\displaystyle=N_{-}^{2}+\delta_{1}^{2}$ $\displaystyle N^{2}$
$\displaystyle=N_{+}^{2}+N_{1}(N_{1}-2N_{-}).$

This gives us the differential equations (using $\Sigma^{2}+N^{2}=1$):

$\displaystyle N_{-}^{\prime}$
$\displaystyle=(N^{2}-1-\Sigma_{+})N_{-}+\sqrt{3}\Sigma_{-}N_{+}$
$\displaystyle N_{+}^{\prime}$
$\displaystyle=(N^{2}-1-\Sigma_{+})N_{+}+\sqrt{3}\Sigma_{-}N_{-}$
$\displaystyle\Sigma_{-}^{\prime}$
$\displaystyle=N^{2}\Sigma_{-}-\sqrt{3}\left(N_{-}N_{+}-N_{+}N_{1}\right),$

allowing us to further compute

$\displaystyle\frac{r_{1}^{\prime}}{r_{1}}$
$\displaystyle=\frac{\Sigma_{-}\Sigma_{-}^{\prime}+N_{-}N_{-}^{\prime}}{r_{1}^{2}}=N^{2}-(\Sigma_{+}+1)\frac{N_{-}^{2}}{r_{1}^{2}}+\sqrt{3}N_{1}\frac{\Sigma_{-}N_{+}}{r_{1}^{2}}$
(2.3.4a) $\displaystyle\psi^{\prime}$
$\displaystyle=\frac{\Sigma_{-}N_{-}^{\prime}-N_{-}\Sigma_{-}^{\prime}}{r_{1}^{2}}=\sqrt{3}N_{+}-(\Sigma_{+}+1)\frac{N_{-}\Sigma_{-}}{r_{1}^{2}}-\sqrt{3}N_{1}\frac{N_{-}N_{+}}{r_{1}^{2}}$
(2.3.4b) $\displaystyle\frac{\delta_{1}^{\prime}}{\delta_{1}}$
$\displaystyle=N^{2}-(\Sigma_{+}+1)$ (2.3.4c)
$\displaystyle\partial_{t}\log\frac{\delta_{1}}{r_{1}}$
$\displaystyle=-(\Sigma_{+}+1)\frac{\Sigma_{-}^{2}}{r_{1}^{2}}-\sqrt{3}N_{1}\frac{\Sigma_{-}N_{+}}{r_{1}^{2}}.$
(2.3.4d)

Near ${\mathbf{T}}_{1}$, i.e. for $\Sigma_{+}\approx-1$, we can use
$1+\Sigma_{+}=\frac{\Sigma_{-}^{2}+N^{2}}{1-\Sigma_{+}}$ in order to rearrange
some terms in (2.3.4):

$\displaystyle\frac{r_{1}^{\prime}}{r_{1}}$
$\displaystyle=\frac{-\Sigma_{+}}{1-\Sigma_{+}}r_{1}^{2}\sin^{2}\psi+\delta_{1}^{2}\frac{\cos^{2}\psi-\Sigma_{+}}{1-\Sigma_{+}}+N_{1}\,h_{r}$
(2.3.5a) $\displaystyle\frac{\delta_{1}^{\prime}}{\delta_{1}}$
$\displaystyle=\frac{-1}{1-\Sigma_{+}}r_{1}^{2}\cos^{2}\psi+\frac{-\Sigma_{+}}{1-\Sigma_{+}}r_{1}^{2}\sin^{2}\psi+\frac{-\Sigma_{+}}{1-\Sigma_{+}}\delta_{1}^{2}+N_{1}\,h_{\delta}$
(2.3.5b) $\displaystyle\partial_{t}\log\frac{\delta_{1}}{r_{1}}$
$\displaystyle=\frac{-1}{1-\Sigma_{+}}r_{1}^{2}\cos^{2}\psi-\delta_{1}^{2}\frac{\cos^{2}\psi}{1-\Sigma_{+}}+N_{1}(h_{\delta}-h_{r})$
(2.3.5c) $\displaystyle\psi^{\prime}$
$\displaystyle=\sqrt{3}r_{1}\sqrt{\cos^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}-\frac{r_{1}^{2}+\delta_{1}^{2}}{1-\Sigma_{+}}\cos\psi\sin\psi+N_{1}\sin\psi\,h_{\psi},$
(2.3.5d)

where

$\displaystyle h_{r}$
$\displaystyle=-\sqrt{3}\frac{\Sigma_{-}N_{+}}{r_{1}^{2}}+(N_{1}-2N_{-})\left(1-\frac{N_{-}^{2}}{(1-\Sigma_{+})r_{1}^{2}}\right)$
(2.3.6a) $\displaystyle h_{\delta}$
$\displaystyle=(N_{1}-2N_{-})\frac{-\Sigma_{+}}{1-\Sigma_{+}}$ (2.3.6b)
$\displaystyle h_{\psi}$
$\displaystyle=-\sqrt{3}\sqrt{\sin^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}-\sin\psi\frac{N_{1}-2N_{-}}{1-\Sigma_{+}}.$
(2.3.6c)

Let us point out that, if $|N_{1}|,|N_{+}|,|N_{-}|\leq\frac{1}{2}$ and
$\Sigma_{+}<0$ and $\frac{\delta_{1}}{r_{1}}\leq 1$, then
$\frac{N_{+}}{r_{1}}\leq\sqrt{2}$ and hence
$|h_{r}|,|h_{\delta}|,|h_{\psi}|\leq 5$.

##### Motivation for polar coordinates around $\mathcal{T}_{1}^{G}$.

One possible motivation for a priori expecting useful equations from this
approach is by a symmetry argument: The Taub-space is invariant since it is
the fixed point space of the reflection
$\sigma:(\Sigma_{+},\Sigma_{-},N_{1},N_{2},N_{3})\to(\Sigma_{+},-\Sigma_{-},N_{1},N_{3},N_{2})$,
and (2.2.2) is equivariant under $\sigma$. This transformation
$\sigma:\mathcal{M}_{\pm+-}\to\mathcal{M}_{\pm-+}$ does not map the quadrant
$N_{2}>0>N_{3}$ into itself. Instead, we have $\mathcal{T}^{G}_{1}$ as the
fixed point space of
$\widetilde{\sigma}:(\Sigma_{+},\Sigma_{-},N_{1},N_{2},N_{3})\to(\Sigma_{+},-\Sigma_{-},N_{1},-N_{3},-N_{2})$.
The Wainwright-Hsu equations are not equivariant under this reflection
$\widetilde{\sigma}$, and we therefore have no reason to expect the fixed-
point space $\mathcal{T}^{G}_{1}$ of $\widetilde{\sigma}$ to be invariant.
However, considering (2.2.2), equivariance is only spoiled by terms of the
form $N_{2}N_{1}$ and $N_{3}N_{1}$ changing their signs; hence, we expect
$\mathcal{T}^{G}_{1}=\\{{\bm{x}}:\,\Sigma_{-}=0,\,|N_{2}|-|N_{3}|=0\\}$ to be
invariant up to terms of order $|N_{2}N_{1}|$ and $|N_{3}N_{1}|$. Such terms
can be well controlled, as it will turn out in Section 5.

##  3 Description of the Dynamics

We will now give an overview of the behaviour of trajectories of (2.2.2). This
overview will contain most of the classic results about Bianchi cosmological
models.

Our overview will be organized by first describing the simplest subsets named
in Table 2 and then progressing to the higher dimensional subsets, finally
describing Bianchi Type IX ($\mathcal{M}_{+++}$) and Bianchi Type VIII
($\mathcal{M}_{+-+}$) solutions. Our approach in this section is very similar
to [Rin01] and [HU09b]; unless explicitly otherwise stated, all observations
in this section can be found therein.

##### A very short summary of relevant dynamics.

The Kasner circle $\mathcal{K}$ is actually a circle and consists entirely of
equilibria. The so-called Mixmaster attractor $\mathcal{A}$ consists of three
$2$-spheres
$\\{{\bm{x}}:\,\Sigma_{+}^{2}+\Sigma_{-}^{2}+N_{i}^{2}=1\,N_{j}=N_{k}=0\\}$,
which intersect in $\mathcal{K}$. Only half of these spheres are accessible
for any trajectory, since the $\mathrm{sign}\,N_{i}$ are fixed. For this
reason, these half-spheres are also called “Kasner-caps”, i.e.
$\mathcal{A}_{+00}$ is the $N_{1}>0$-cap.

The dynamics on the Kasner-caps will be discussed in Section 3.1; each orbit
in a Kasner-cap is a heteroclinic orbit connecting two equilibria on the
Kasner-circle $\mathcal{K}$.

The long-time behaviour of the lower dimensional Bianchi-types (at least one
$N_{i}=0$) is well-understood: All such solutions converge to an equilibrium
${\bm{p}}\in\mathcal{K}$ as $t\to\infty$. The behavior in the highest-
dimensional Bianchi Types IX and VIII is not yet fully understood, and is
therefore of most interest in this work.

It is known (c.f. [Rin01]) that Bianchi Type IX solutions that do not lie in a
Taub-space converge to the Mixmaster attractor as $t\to+\infty$, i.e. towards
the big bang singularity. It has been conjectured that generic Bianchi Type
VIII solutions share this behaviour; this will be proven in this work (Theorem
4 and 5).

The question of particle horizons was already mentioned in the introduction
and is further discussed from a physical viewpoint in Section 8. In terms of
the Wainwright-Hsu equations, the question can be formulated as (see Section
5, or c.f. e.g. [HR09]):

$\text{Is }\quad
I({\bm{x}})=\max_{i}\int_{0}^{\infty}\delta_{i}(\phi({\bm{x}},t)){\mathrm{d}}t=2\max_{i}\int_{0}^{\infty}\sqrt{|N_{j}N_{k}|}(t){\mathrm{d}}t<\infty\,?$

Here $\phi$ is the flow to (2.2.2). The space-time associated to the solution
$\phi({\bm{x}},\cdot)$ forms a particle horizon if and only if $I<\infty$.

It is known that there exist solutions in Bianchi IX and VIII, where
$I<\infty$ (c.f. [LHWG11]). It is not known, whether there exist any
nontrivial solutions with $I=\infty$ (of course solutions starting in
$\mathcal{T}$ that do not converge to $\mathcal{A}$ must have $I=\infty$). We
prove that, in both Bianchi IX and VIII and for Lebesgue almost every initial
condition, particle horizons develop ($I<\infty$) (Theorem 6).

###  3.1 Lower Bianchi Types

##### Bianchi Type I: The Kasner circle.

The smallest, i.e. lowest dimensional, Bianchi-type is Type I,
$\mathcal{M}_{000}=\mathcal{K}$, where all three $N_{i}$ vanish (see Table 2).
By the constraint $N^{2}+\Sigma^{2}=1$, we can see that $\mathcal{K}$ is the
unit circle in the $(\Sigma_{+},\Sigma_{-})$-plane, and consists entirely of
equilibria.

The linear stability of these equilibria is given by the following

######  Lemma 3.1.

Let ${\bm{p}}=(\Sigma_{+},\Sigma_{-},0,0,0)\in\mathcal{K}$; first consider the
case ${\bm{p}}\neq{\mathbf{T}}_{i}$ and without loss of generality
$|{\bm{p}}+{\mathbf{T}}_{1}|<1$. Then the vectorfield has one central
direction given by $\partial_{\mathcal{K}}=(-\Sigma_{-},\Sigma_{+},0,0,0)$,
one unstable direction given by $\partial_{N_{1}}=(0,0,1,0,0)$, and two stable
directions given by $\partial_{N_{2}}=(0,0,0,1,0)$ and
$\partial_{N_{3}}=(0,0,0,0,1)$.

The three Taub-points ${\mathbf{T}}_{i}$ have each one stable direction given
by $\partial_{N_{i}}$ and three center directions given by
$\partial_{\mathcal{K}}$, $\partial_{N_{j}}$ and $\partial_{N_{k}}$.

###### Proof.

We first note that the four vectors $\partial_{\mathcal{K}}$ and
$\partial_{N_{i}}$ form a basis of the tangent space
$T_{{\bm{p}}}\mathcal{M}=\mathrm{ker}\,{\mathrm{d}}G$.

The stability of an equilibrium is determined by the eigenvalues and
eigenspaces of the Jacobian of the vector field; a generalized eigenspace is
_central_ , if its eigenvalue has vanishing real part, it is _stable_ if its
eigenvalue has negative real part, and it is _unstable_ if its eigenvalue has
positive real part. The Jacobian ${\mathrm{D}}f$ of the vector field $f$ given
by (2.2.2) at ${\bm{p}}\in\mathcal{K}$ is diagonal with three entries of the
form
${\mathrm{D}}f=\lambda_{1}\partial_{N_{1}}\otimes{\mathrm{d}}N_{1}+\lambda_{N_{2}}\partial_{2}\otimes{\mathrm{d}}N_{2}+\lambda_{3}\partial_{N_{3}}\otimes{\mathrm{d}}N_{3}$;
we can read off the stability from (2.2.2b) and Figure 2(a). ∎

(a) The three discs, where $N_{i}^{\prime}/N_{i}>0$ and a short heteroclinic
chain.

(b) The three discs, where $\delta_{i}^{\prime}/\delta_{i}>0$. Also, the
Kasner-map $K$ is a double cover.

Figure 2: Stability properties and the Kasner map. All figures are in
$(\Sigma_{+},\Sigma_{-})$-projection.

##### The Taub-line.

There exists another structure of equilibria, given by
$\mathcal{TL}_{i}:=\\{{\bm{x}}\in\mathcal{M}:N_{j}=N_{k},\,N_{i}=0,\,{\bm{\Sigma}}={\mathbf{T}}_{i}\\}$.
Up to index permutations and ${\bm{N}}\to-{\bm{N}}$, this set has the form
$\mathcal{TL}_{1}\cup\mathcal{M}_{++0}=\\{{\bm{p}}:\,{\bm{p}}=(-1,0,0,n,n),\,n>0\\}$.
We observe that this is a line of equilibria. Each such equilibrium has one
stable direction (corresponding to $N_{1}$) and three center directions.

##### Bianchi Type II: The Kasner caps.

Consider without loss of generality the set $\mathcal{M}_{+00}$. The
constraint $G=1$ then reads

$1=N_{1}^{2}+\Sigma_{+}^{2}+\Sigma_{-}^{2},$

i.e. the so-called “Kasner-cap” $\mathcal{M}_{+00}$ forms a half-sphere with
$\mathcal{K}$ as its boundary. Considering (2.2.2), we can see that
${\bm{\Sigma}}^{\prime}=({\bm{\Sigma}}+2{\mathbf{T}}_{1})N_{1}^{2}$ is a
scalar multiple of ${\bm{\Sigma}}+2{\mathbf{T}}_{1}$; hence, the
${\bm{\Sigma}}$-projection of the trajectory stays on the same line through
$-2{\mathbf{T}}_{1}$. Since $N_{1}^{2}\geq 0$, this trajectory is heteroclinic
and converges in forward and backward times to the two intersections of this
line with the Kasner circle $\mathcal{K}$, where the $\alpha$-limit is closer
to $-2{\mathbf{T}}_{1}$. Such a trajectory is depicted in Figure 2(a).

##### The Mixmaster-Attractor $\mathcal{A}$.

The most relevant set for the long-time behavior of the Bianchi system is the
Mixmaster attractor $\mathcal{A}$. It consists of the union of all six Bianchi
II pieces and the Kasner circle. Since the signs of the $N_{i}$ stay constant
along trajectories, it is useful to only study a piece of $\mathcal{A}$, given
without loss of generality by:

$\mathcal{A}_{+++}=\mathcal{M}_{+00}\cup\mathcal{M}_{0+0}\cup\mathcal{M}_{00+}\cup\mathcal{K}.$

The set $\mathcal{A}_{+++}$ is given by the union of three perpendicular half-
spheres (the three Kasner-caps), which intersect in the Kasner-circle. The
dynamics in the Mixmaster-Attractor now consists of the Kasner circle
$\mathcal{K}$ of equilibria and the three caps, consisting of heteroclinic
orbits to $\mathcal{K}$. It is described in detail by the so-called Kasner-
map.

##### The Kasner-map $K:\mathcal{K}\to\mathcal{K}$.

We wish to describe which equilibria in $\mathcal{K}$ are connected by
heteroclinic orbits. We can collect this in a relation
$K\subseteq\mathcal{K}\times\mathcal{K}$, i.e. we write
${\bm{p}}_{-}K{\bm{p}}_{+}$ if either
${\bm{p}}_{-}={\bm{p}}_{+}={\mathbf{T}}_{i}$ or there exists a heteroclinic
orbit $\gamma:{\mathbb{R}}\to\mathcal{A}$ such that
${\bm{p}}_{-}=\lim_{t\to-\infty}\gamma(t)$ and
${\bm{p}}_{+}=\lim_{t\to+\infty}\gamma(t)$.

We can see by Figure 2(a) (or Lemma 3.1) that each non-Taub point
${\bm{p}}_{-}$ has a one-dimensional unstable manifold, i.e. one trajectory in
$\mathcal{A}_{{{\hat{n}}}}$, which converges to ${\bm{p}}_{-}$ in backwards
time. Therefore, the relation $K$ can be considered as a (single-valued,
everywhere defined) map.

This map is depicted in Figure 2(b), and has a simple geometric description in
the $(\Sigma_{+},\Sigma_{-})$-projection: Given some
${\bm{p}}_{-}\in\mathcal{K}$, we draw a straight line through ${\bm{p}}_{-}$
and the nearest of the three points $-2{\mathbf{T}}_{i}$. This line has
typically two intersections ${\bm{p}}_{-}$ and ${\bm{p}}_{+}\in\mathcal{K}$
with the Kasner-circle, one of which is nearer to $-2{\mathbf{T}}_{i}$, which
is ${\bm{p}}_{-}$, and one which is further away, which is ${\bm{p}}_{+}$. At
a ${\mathbf{T}}_{i}$, there are two possible choices of nearest
$-2{\mathbf{T}}_{j}$ and $-2{\mathbf{T}}_{k}$ and the lines through these
points are tangent to $\mathcal{K}$; we just set
$K({\mathbf{T}}_{i})={\mathbf{T}}_{i}$. We see from Figure 2(b) that this map
$K:\mathcal{K}\to\mathcal{K}$ is continuous and a double cover (i.e. each
point ${\bm{p}}_{+}$ has two preimages ${\bm{p}}_{-}^{1}$ and
${\bm{p}}_{-}^{2}$, which both depend continuously on ${\bm{p}}_{+}$).

Looking at Figure 2(a), we can also see that the Kasner-map is expanding.
Hence it is $C^{0}$-conjugate to either
$[z]_{{\mathbb{Z}}}\to[2z]_{{\mathbb{Z}}}$ or
$[z]_{{\mathbb{Z}}}\to[-2z]_{{\mathbb{Z}}}$; since it has three fixed points
the latter case must apply. Hence we have

######  Proposition 3.2.

There exists a homeomorphism $\psi:\mathcal{K}\to{\mathbb{R}}/3{\mathbb{Z}}$,
such that $\psi({\mathbf{T}}_{i})=\left[i\right]_{3{\mathbb{Z}}}$ and

$\psi(K({\bm{p}}))=[-2\psi({\bm{p}})]_{3{\mathbb{Z}}}\qquad\forall{\bm{p}}\in\mathcal{K}.$

A formal proof of Proposition 3.2 is a digression; for this reason, it is
deferred until Section A.2, where we give a more detailed description of the
Kasner map.

##### Basic Heuristics near the Mixmaster Attractor.

Heuristically, the Kasner-map determines the behavior of solutions near
$\mathcal{A}$: Consider an initial condition
${\bm{x}}_{0}\in\mathcal{M}_{\pm\pm\pm}$ near $\mathcal{A}$, i.e. an initial
condition where none of the $N_{i}$ vanish. Then the trajectory ${\bm{x}}(t)$
will closely follow the heteroclinic solution $\gamma_{1}$ passing near
${\bm{x}}_{0}$. Let ${\bm{p}}_{1}$ be the end-point of this heteroclinic;
${\bm{x}}(t)$ will follow $\gamma_{1}$ and stay for some time near
${\bm{p}}_{1}$ (since it is an equilibrium). However, if
${\bm{p}}_{1}\neq{\mathbf{T}}_{i}$, then one of the $N_{i}$ directions is
unstable; therefore, ${\bm{x}}(t)$ will leave the neighborhood of
${\bm{p}}_{1}$ along the unique heteroclinic emanating from ${\bm{p}}_{1}$,
and follow it until it is near ${\bm{p}}_{2}=K({\bm{p}}_{1})$. This should
continue until ${\bm{x}}(t)$ leaves the vicinity of $\mathcal{A}$, which
should not happen at all (at least if the name “Mixmaster Attractor” is well
deserved).

The expansion of the Kasner-map is the source of the (so far heuristic)
chaoticity of the dynamics of the Bianchi system: The expansion along the
Kasner-circle supplies the sensitive dependence on initial conditions, while
the remaining two directions are contracting. Since we study a flow in four-
dimensions, Lyapunov exponents need only account for three dimensions, the
last one corresponding to the time-evolution.

##### Bianchi-Types $\textsc{VII}_{0}$ and $\textsc{VI}_{0}$.

There are two Bianchi-types, where exactly one of the three $N_{i}$ vanishes:
Types $\textsc{VII}_{0}$ and $\textsc{VI}_{0}$. In these Bianchi-Types, we
have monotone functions (Lyapunov functions), which suffice to almost
completely determine the long-time behaviour of trajectories. Without loss of
generality, we focus on the case where $N_{1}=0$. Then we can write

$\Sigma_{+}^{\prime}=(1-\Sigma^{2})(\Sigma_{+}+1)\qquad
1=\Sigma^{2}+(N_{2}-N_{3})^{2}.$

We can immediately see that $\Sigma_{+}$ is non-decreasing along trajectories;
indeed, we must have $\lim_{t\to\pm\infty}\Sigma^{2}(t)=1$ for all
trajectories. Considering (2.2.2b) and Figure 2(a), we can see that, for
$t\to+\infty$ we must either have $\Sigma_{+}\geq\frac{1}{2}$, since both
$N_{2}$ and $N_{3}$ must be stable, or $\Sigma_{+}=-1$ all along; then
$N_{2}=N_{3}$ and the trajectory lies in the Taub-line $\mathcal{TL}_{1}$. In
backward time, we must have $\Sigma_{+}\to-1$: All other points on the Kasner-
circle have one of the two $N_{2},N_{3}$ unstable. These statements can be
formalized as

######  Lemma 3.3.

Consider an initial condition ${\bm{x}}_{0}$ with
${\bm{x}}_{0}\in\mathcal{M}_{0-+}$. Then, in forward time, the trajectory
${\bm{x}}(t)=\phi({\bm{x}}_{0},t)$ converges to a point on the Kasner-circle
$\lim_{t\to\infty}{\bm{x}}(t)={\bm{p}}_{+}\in\mathcal{K}$ with
$\Sigma_{+}({\bm{p}}_{+})\geq\frac{1}{2}$. In backwards time, the trajectory
converges to the Taub-point
${\mathbf{T}}_{1}=(-1,0,0,0,0)=\lim_{t\to-\infty}{\bm{x}}(t)$.

######  Lemma 3.4.

Consider an initial condition ${\bm{x}}_{0}$ with
${\bm{x}}_{0}\in\mathcal{M}_{0++}\setminus\mathcal{TL}_{1}$. Then, in forward
time, the trajectory ${\bm{x}}(t)=\phi({\bm{x}}_{0},t)$ converges to a point
on the Kasner-circle $\lim_{t\to\infty}{\bm{x}}(t)={\bm{p}}_{+}\in\mathcal{K}$
with $\Sigma_{+}({\bm{p}}_{+})\geq\frac{1}{2}$. In backwards time, the
${\bm{\Sigma}}$-projection of the trajectory converges to the Taub-point
${\mathbf{T}}_{1}=(-1,0)=\lim_{t\to-\infty}{\bm{\Sigma}}({\bm{x}}(t))$. No
claim about the dynamics of the $N_{i}(t)$ for $t\to-\infty$ is made.

###### Proof.

Contained in the preceding paragraph. ∎

It can be shown that, in the case of Bianchi $\textsc{VII}_{0}$, i.e.
${\bm{x}}_{0}\in\mathcal{M}_{0++}\setminus\mathcal{TL}_{1}$, the limit
$\lim_{t\to-\infty}{\bm{x}}(t)={\bm{p}}\in\mathcal{TL}_{1}\setminus\\{{\mathbf{T}}_{1}\\}$
exists and does not lie on the Kasner circle. This claim follows directly from
Lemma 5.6; however, the proof of Lemma 5.6 is rather lengthy and not required
for our main results.

###  3.2 Bianchi-Types VIII and IX for large $N$

As we have seen, the lower Bianchi types do not support recurrent dynamics.
This is different in the two top-dimensional Bianchi-types VIII and IX. This
section is devoted to describing the behaviour far from $\mathcal{A}$.

######  Lemma 3.5 (Long-Time Existence).

Every solution ${\bm{x}}:[0,T)\to\mathcal{M}$ of $\eqref{eq:ode}$ has bounded
$\Sigma^{2}(t)<C({\bm{x}}_{0})$ for all $t\in[0,T)$ and has unbounded forward
existence time (i.e. no finite-time blow-up occurs towards the future, i.e.
towards the big bang singularity).

The product $|N_{1}N_{2}N_{3}|$ is non-increasing along solutions
${\bm{x}}(t)$, since (from (2.2.4)):

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\log|N_{1}N_{2}N_{3}|=-3\Sigma^{2}\leq 0.$

###### Proof.

The monotonicity of $|N_{1}N_{2}N_{3}|$ is already proven in the statement.

The only way that long-term existence can fail is finite-time blow-up, i.e.
$\lim_{t\to T_{\max}}|{\bm{x}}(t)|=\infty$ for some $0<T_{\max}<\infty$. We
cannot exclude this possibility a priori, since the vectorfield given by
(2.2.2) is polynomial. However, it suffices to estimate
$|{\bm{x}}^{\prime}(t)|\leq C+C|{\bm{x}}|$, with constants independent of of
$t\geq 0$ (but possibly depending on ${\bm{x}}_{0}$).

We first consider the case of Bianchi Type VIII, without loss of generality
$\mathcal{M}_{-++}$. Consider a maximal solution
${\bm{x}}:[0,T_{\max})\to\mathcal{M}_{-++}$. Since
$N^{2}=(N_{1}-N_{2}+N_{3})^{2}-4N_{1}N_{3}>0$, we can see that $N^{2}>0$ and
(from $1=\Sigma^{2}+N^{2}$) that $\Sigma^{2}<1$ for all times $t<T_{\max}$.
From this, we can easily estimate $|{\bm{x}}^{\prime}|\leq C+C|{\bm{x}}|$;
therefore, solutions exist for all positive times (i.e. $T_{\max}=\infty$).

Next, we consider the case of Bianchi Type IX, without loss of generality
${\bm{x}}:[0,T_{\max})\to\mathcal{M}_{+++}$. If $N_{2}\geq\max(N_{1},N_{3})$,
then we can see

$N^{2}=(N_{1}-N_{2}+N_{3})^{2}-4N_{1}N_{3}\geq-4N_{1}N_{3}\geq-4\left(N_{1}N_{2}N_{3}\right)^{\frac{2}{3}}\left(\frac{N_{1}N_{3}}{N_{2}N_{2}}\right)^{\frac{1}{3}}\geq-4\left(N_{1}N_{2}N_{3}\right)^{\frac{2}{3}}.$

By permutation symmetry, the above inequality holds regardless of which
$N_{i}$ is largest. Therefore, we have for all times $t>0$:

$N^{2}(t)\geq-4\left(N_{1}N_{2}N_{3}\right)^{\frac{2}{3}}(0),\qquad\Sigma^{2}(t)=1-N^{2}(t)\leq
1+4\left(N_{1}N_{2}N_{3}\right)^{\frac{2}{3}}(0).$

Unbounded time of existence follows as in the case of Bianchi VIII. ∎

Next, we show that $|N_{1}N_{2}N_{3}|\to 0$ as $t\to\infty$, and that this
convergence is essentially uniformly exponential:

######  Lemma 3.6 (Essentially exponential convergence of
$|N_{1}N_{2}N_{3}|$).

For every $C_{N}>0$, there exist constants $C_{1},C_{0}>0$ such that for all
trajectories ${\bm{x}}:[t_{1},t_{2}]\to\mathcal{M}$ with
$|N_{1}N_{2}N_{3}|(t_{1})<C_{N}$ we have

$|N_{1}N_{2}N_{3}|(t_{2})\leq|N_{1}N_{2}N_{3}|(t_{1})\exp\left(C_{0}-C_{1}(t_{2}-t_{1})\right).$

###### Proof.

Consider the function $s:\mathcal{M}\to[0,\infty)$

$s({\bm{x}})=\int_{0}^{1}\Sigma^{2}(\phi({\bm{x}},t)){\mathrm{d}}t,$

where $\phi:\mathcal{M}\times{\mathbb{R}}\to\mathcal{M}$ is the flow
associated to (2.2.2). Fix some $C_{N}>0$. We will show that we find some
constant $\widetilde{C}>0$ such that $s({\bm{x}})>\widetilde{C}$ whenever
$|N_{1}N_{2}N_{3}|({\bm{x}})\leq C_{N}$. From this, we can conclude

$\log\frac{|N_{1}N_{2}N_{3}|(t_{2})}{|N_{1}N_{2}N_{3}|(t_{1})}=-3\int_{t_{1}}^{t_{2}}\Sigma^{2}(t){\mathrm{d}}t\leq-3(s(t_{1})+s(t_{1}+1)+\ldots)\leq-3(t_{2}-t_{1}-1)\widetilde{C}.$

We will prove the estimate $s({\bm{x}})>\widetilde{C}(C_{N})$ by
contradiction. Assume that we had a sequence $\\{{\bm{x}}_{n}\\}$, such that
$s({\bm{x}}_{n})\to 0$ and $|N_{1}N_{2}N_{3}|({\bm{x}}_{n})$ is bounded as
$n\to\infty$.

At first, we show that this cannot happen in bounded regions of phase-space:
We can see from (2.2.2) that there do not exist any invariant sets with
$\Sigma^{2}=0$ (because then $N^{2}=1$ and $\Sigma^{\prime}\neq 0$). Therefore
$s({\bm{x}})>0$ for all ${\bm{x}}\in\mathcal{M}$. Since $s$ is continuous, the
sequence ${\bm{x}}_{n}$ cannot converge, and hence cannot be bounded
(otherwise, there would be some convergent subsequence).

We can assume without loss of generality that $\Sigma^{2}({\bm{x}}_{n})\to 0$
(since we assumed
$\int_{0}^{1}\Sigma^{2}(\phi({\bm{x}}_{n},t)){\mathrm{d}}t\to 0$). In order to
avoid convergent subsequences, we must have
$\max_{i}|N_{i}|({\bm{x}}_{n})\to\infty$ as $n\to\infty$.

Consider first the case of Bianchi VIII with ${\bm{x}}\in\mathcal{M}_{-++}$.
Then

$N^{2}=(N_{2}-N_{3})^{2}+N_{1}^{2}-2N_{1}(N_{2}+N_{3})=1-\Sigma^{2}.$

All three terms in the middle are non-negative and hence bounded; therefore,
we must have $|N_{2}|,|N_{3}|\to\infty$ and $|N_{1}|\to 0$. We can write

${\mathrm{D}}_{t}\Sigma_{+}=(1-\Sigma^{2})(\Sigma_{+}+1)+3N_{1}(N_{2}+N_{3}-N_{1})$

We can estimate the terms involving $N_{i}$ as

$|N_{1}|(N_{2}+N_{3}+|N_{1}|)\leq\frac{C_{N}}{\max(N_{2},N_{3})}\to
0\qquad\text{as $n\to\infty$}.$

Since ${\mathrm{D}}_{t}\log{|N_{i}|}\leq|{\bm{\Sigma}}|(2+|{\bm{\Sigma}}|)$ is
bounded, the estimate $N_{2},N_{3}\gg 1\gg|N_{1}|$ and hence
$|N_{1}|(N_{2}+N_{3}+|N_{1}|)\ll 1$ stay valid for at least one unit of time.
We therefore cannot have $\lim_{n\to\infty}s({\bm{x}}_{n})=0$:
$s({\bm{x}}_{n})\approx 0$ is only possible if
$\max_{t\in[0,1]}\Sigma^{2}(\phi({\bm{x}}_{n},t))\approx 0$. This is, however,
impossible since then $\Sigma_{+}^{\prime}\approx 1$.

Consider now the case of Bianchi IX, i.e. ${\bm{x}}\in\mathcal{M}_{+++}$.
Assume without loss of generality that $N_{3}\geq N_{2}\geq N_{1}$. Then we
can write

$1-\Sigma^{2}=N^{2}=(N_{1}+N_{2}-N_{3})^{2}-4N_{1}N_{2}\geq(N_{1}+N_{2}-N_{3})^{2}-4C_{N}^{\frac{2}{3}}.$

Therefore, we must have $N_{2},N_{3}\to\infty$ and $N_{1}\to 0$. Apart from
this, the same arguments as for Bianchi VIII apply. ∎

This result, i.e. Lemma 3.6, is not as explicitly stated in the previous works
[Rin01, HU09b], and certainly not as extensively used, but is not a novel
insight either. It directly proves that metric coefficients stay bounded, see
Section 8.

Using Lemma 3.6, we can quickly see the following:

######  Lemma 3.7 (Existence of $\omega$-limits).

For any initial condition ${\bm{x}}_{0}\in\mathcal{M}$, the $\omega$-limit set
is nonempty, $\omega({\bm{x}}_{0})\neq\emptyset$, i.e. there exists a sequence
of times $(t_{n})_{n\in{\mathbb{N}}}$ with $\lim_{n\to\infty}t_{n}=\infty$
such that the limit $\lim_{n\to\infty}{\bm{x}}(t_{n})$ exists.

###### Proof.

We begin again by considering the Bianchi VIII case of
${\bm{x}}_{0}\in\mathcal{M}_{-++}$. The only way of avoiding the existence of
an $\omega$-limit is to have $\lim_{t\to\infty}|{\bm{x}}(t)|=\infty$. As in
the proof of Lemma 3.6, this is only possible via $N_{2},N_{3}\to\infty$ and
$N_{1}\to 0$. Then

$\displaystyle\Sigma_{+}^{\prime}(t)$
$\displaystyle=(1-\Sigma^{2})(\Sigma_{+}+1)+3N_{1}(N_{2}+N_{3}-N_{1})$
$\displaystyle\geq(1-\Sigma^{2})(\Sigma_{+}+1)-9|N_{1}N_{2}N_{3}|$
$\displaystyle\geq(1-\Sigma^{2})(\Sigma_{+}+1)-C_{1}e^{-C_{2}t}$
$\displaystyle\frac{\delta_{1}^{\prime}}{\delta_{1}}(t)$
$\displaystyle=-(\Sigma^{2}+\Sigma_{+}),$

where $\delta_{1}=2\sqrt{|N_{2}N_{3}|}$ as in (2.2.7) and (2.2.8). There are
basically two possibilities with regards to the dynamics of $\Sigma_{+}$: If
$\Sigma_{+}\to-1$ as $t\to+\infty$, then this convergence must happen
exponentially, since $(1-\Sigma^{2})(\Sigma_{+}+1)\geq 0$. That is, if
$\Sigma_{+}\to-1$, then we must have
$|\Sigma_{+}+1|(t)\leq\frac{C_{1}}{C_{2}}e^{-C_{2}t}$ for all sufficiently
large times $t>t_{0}>0$. Then

$\int_{t_{0}}^{\infty}\partial_{t}\log\delta_{1}(t){\mathrm{d}}t=\int_{0}^{\infty}-\Sigma_{-}^{2}-\Sigma_{+}(1+\Sigma_{+}){\mathrm{d}}t\leq
2\frac{C_{1}}{C_{2}^{2}}e^{-C_{2}t_{0}}<\infty,$

and hence $\lim_{t\to\infty}\delta_{1}(t)<\infty$, contradicting our
assumption.

The other option is to have $1+\Sigma_{+}\not\to-1$ as $t\to\infty$. Then we
must have $1+\Sigma_{+}(t)>\epsilon$ for some $\epsilon>0$ for all
sufficiently large times. Informally, we can see from Figure 2(b) that this
contradicts $\delta_{1}\to\infty$. Formally, we can say: Since $\Sigma_{+}$ is
bounded,
$\int(1-\Sigma^{2})\epsilon{\mathrm{d}}t<\int\Sigma_{+}^{\prime}{\mathrm{d}}t+C<\infty$,
and hence $\int(1-\Sigma^{2}){\mathrm{d}}t<\infty$. On the other hand,
${\mathrm{D}}_{t}\log(\delta_{1})=-\Sigma^{2}-\Sigma_{+}\leq
1-\Sigma^{2}-\epsilon$ and integration shows
$\lim_{t\to\infty}\delta_{1}(t)=0$, contradicting our assumption
$N_{2},N_{3}\to\infty$.

Next, we consider the Bianchi IX case of ${\bm{x}}_{0}\in\mathcal{M}_{+++}$.
Again, the only way of avoiding the existence of an $\omega$-limit is to have
$\lim_{t\to\infty}|{\bm{x}}(t)|=\infty$. As in the proof of Lemma 3.6, this is
only possible via $N_{2},N_{3}\to\infty$ and $N_{1}\to 0$ for some permutation
of indices. Using
$1-\Sigma^{2}\geq|1-\Sigma^{2}|-8|N_{1}N_{2}N_{3}|^{\frac{2}{3}}$ and
replacing $1-\Sigma^{2}$ by $|1-\Sigma^{2}|$, we can use the same arguments as
in the Bianchi VIII case. ∎

###  3.3 The Bianchi IX Attractor Theorem

The Mixmaster Attractor was named in the 60s. However, the first proof that
$\mathcal{A}$ actually is an attractor was given in [Rin01, Theorem $19.2$,
page 65], and simplified in [HU09b]. We shall state this important result:

######  Theorem 1 (Classical Bianchi IX Attractor Theorem).

Let ${\bm{x}}_{0}\in\mathcal{M}_{+++}\setminus\mathcal{T}$. Then

$\lim_{t\to\infty}\mathrm{dist}({\bm{x}}(t),\mathcal{A})=0.$

Also, the $\omega$-limit set $\omega({\bm{x}}_{0})$ does not consist of a
single point.

The proofs of Theorem 1 given in [Rin01] and [HU09b] require some subtle
averaging arguments (summarized as Lemma 5.6), which are lengthy and fail to
generalize to the case of Bianchi VIII initial data. We will now give the
first steps leading to the proof of Theorem 1, up to the missing averaging
estimates for Bianchi IX solutions. Then, we will state the missing estimates
and show how they prove Theorem 1. Afterwards, we will give a high-level
overview of how we replace Lemma 5.6 in this work. Nevertheless, for the sake
of completeness, we provide a proof of Lemma 5.6 in Section 5.4.

##### Rigorous steps leading to Theorem 1.

We first show that solutions cannot converge to the Taub-line
$\mathcal{TL}_{i}$, if they do not start in the Taub-space $\mathcal{T}_{i}$:

######  Lemma 3.8 (Taub Space Instability).

Let $\epsilon>0$ small enough. Then there exists a constant
$C_{r,\epsilon}\in(0,1)$, such that the following holds:

Recall the definition of $r_{1}$, which measures the distance to
$\mathcal{T}_{1}$ (see (2.2.7)). For any piece of trajectory
${\bm{x}}:[t_{1},t_{2}]\to\\{{\bm{x}}\in\mathcal{M}_{*++}:|{\bm{\Sigma}}({\bm{x}})-{\mathbf{T}}_{1}|\leq\epsilon\\}$,
the following estimate holds:

$\displaystyle r_{1}(\gamma(t_{2}))$ $\displaystyle\geq
C_{r,\epsilon}h(\gamma(t_{1}))r_{1}(\gamma(t_{1})),\qquad\text{ where}$
$\displaystyle h({\bm{x}})$
$\displaystyle=|N_{1}|+|N_{1}|^{2}+|N_{1}N_{2}N_{3}|.$

###### Proof.

This Lemma uses the polar coordinates introduced in Section 2.3. We use
(2.3.2) to see

$\displaystyle{\mathrm{D}}_{t}\log r_{1}$ $\displaystyle\geq
r_{1}^{2}\sin^{2}\psi\frac{-\Sigma_{+}}{1-\Sigma_{+}}-C|N_{1}|-C|N_{1}^{2}|-C|N_{1}N_{2}N_{3}|,$
$\displaystyle{\mathrm{D}}_{t}\log|N_{1}|$ $\displaystyle\approx-3<-1,$
$\displaystyle{\mathrm{D}}_{t}\log|N_{1}N_{2}N_{3}|$
$\displaystyle\approx-3<-1.$

The desired estimate follows by integration. ∎

Together with Lemma 3.7, this allows us to see that there exist $\omega$-limit
points on $\mathcal{K}$:

######  Lemma 3.9.

Let ${\bm{x}}_{0}\in\mathcal{M}_{+++}\setminus\mathcal{T}$. Then there exists
at least one $\omega$-limit point
${\bm{p}}\in\left(\mathcal{K}\setminus\\{{\mathbf{T}}_{1},{\mathbf{T}}_{2},{\mathbf{T}}_{3}\\}\right)\cap\omega({\bm{x}}_{0})$.

Let ${\bm{x}}_{0}\in\mathcal{M}_{+-+}\setminus\mathcal{T}_{2}$. Then there
exists at least one $\omega$-limit point
${\bm{p}}\in\left(\mathcal{K}\setminus\\{{\mathbf{T}}_{2}\\}\right)\cap\omega({\bm{x}}_{0})$.

###### Proof.

We already know that there exists an $\omega$-limit point
${\bm{y}}\in\omega({\bm{x}}_{0})$; this point must have $|N_{1}N_{2}N_{3}|=0$
and hence be of a lower Bianchi type. In view of Lemma 3.3 and Lemma 3.4 and
the fact that both $\alpha({\bm{y}})\subseteq\omega({\bm{x}}_{0})$ and
$\omega({\bm{y}})\subseteq\omega({\bm{x}}_{0})$, it suffices to exclude the
case where
$\omega({\bm{x}}_{0})\subseteq\mathcal{T}\mathcal{L}_{i}\setminus\mathcal{K}$
for some $i$. This possibility is excluded by Lemma 3.8. ∎

Therefore, we know that $\liminf_{t\to\infty}\max_{i}\delta_{i}(t)=0$ and
hence $\liminf_{t\to\infty}\mathrm{dist}({\bm{x}}(t),\mathcal{A})=0$ for
initial conditions in $\mathcal{M}_{\pm\pm\pm}\setminus\mathcal{T}$. While we
presently lack the necessary estimates to prove the missing part of the
attractor theorem, $\limsup_{t\to\infty}\max_{i}\delta_{i}(t)=0$, we can at
least describe how this may fail: Each $\delta_{i}$ can only grow by a
meaningful factor in the vicinity of a Taub-point ${\mathbf{T}}_{i}$:

######  Lemma 3.10.

There exists a constant $C>0$ such that, given $\epsilon_{+}>0$, we find
$\epsilon_{123}=\epsilon_{123}(\epsilon_{+})>0$, the following holds:

Suppose we have a piece of trajectory
${\bm{x}}:[t_{1},t_{2}]\to\mathcal{M}_{\pm\pm\pm}$, such that:

  1. 1.

We have the product bound $|N_{1}N_{2}N_{3}|(t)<\epsilon_{123}$ for all
$t\in[t_{1},t_{2}]$

  2. 2.

The first point of the partial trajectory is bounded away from the Taub-line
$\mathcal{TL}_{1}$, i.e. $1+\Sigma_{+}(t_{1})>\epsilon_{+}$

  3. 3.

The piece of trajectory has comparatively large $\delta_{1}$, in the sense
$1\geq\delta_{1}^{4}(t)\geq|N_{1}N_{2}N_{3}|(t)$ for all $t\in[t_{1},t_{2}]$,
i.e. $|N_{1}|\leq 4\delta_{1}^{2}=16|N_{2}N_{3}|$ for all $t\in[t_{1},t_{2}]$.

Then $\delta_{1}$ can only increase by a bounded factor along this piece of
trajectory, i.e.

$\delta_{1}(t_{2})\leq\exp\left(\frac{C}{\epsilon_{+}}\right)\delta_{1}(t_{1}).$

###### Proof.

Recall the proof of Lemma 3.7. From $\delta_{1}^{4}\geq|N_{1}N_{2}N_{3}|$, we
know that $|N_{1}|\leq\delta_{1}^{2}$ and therefore

$\displaystyle\Sigma_{+}^{\prime}$
$\displaystyle=(1-\Sigma^{2})(\Sigma_{+}+1)+3N_{1}(N_{2}+N_{3}-N_{1})$
$\displaystyle\geq|1-\Sigma^{2}|(\Sigma_{+}+1)-C\sqrt{|N_{1}N_{2}N_{3}|}.$

If $\epsilon_{123}>0$ is small enough, this allows us to see that
$1+\Sigma_{+}(t)>\frac{1}{2}\epsilon_{+}$ for all $t\in[t_{0},t_{1}]$. Since
$|\Sigma_{+}|\leq 2$ is bounded, we see that
$\int|1-\Sigma^{2}|\epsilon_{+}{\mathrm{d}}t<C$ and hence
$\int|1-\Sigma^{2}|{\mathrm{d}}t<\frac{C}{\epsilon_{+}}$. On the other hand,
${\mathrm{D}}_{t}\log\delta_{1}<(1-\Sigma^{2})-\frac{1}{2}\epsilon_{+}$, which
yields the claim upon integration. ∎

##### Sketch of classic proofs of Theorem 1.

The previous proofs of Theorem 1, both in [Rin01] and [HU09b], rely on the
following estimate (Lemma $3.1$ in [HU09b], Section $15$ in [Rin01]):

######  Lemma 5.6.

We consider without loss of generality the neighborhood of $\mathcal{T}_{1}$.
Let $\epsilon>0$ small enough. Then there exists a constant
$C_{\delta,\epsilon}\in(1,\infty)$, such that, for any piece of trajectory
$\gamma:[t_{1},t_{2}]\to\\{{\bm{x}}\in\mathcal{M}_{*++}:|{\bm{\Sigma}}({\bm{x}})-{\mathbf{T}}_{1}|\leq\epsilon,\,|N_{1}|\leq
10,|\delta_{1}|\leq 10\\}$, the following estimate holds:

$\delta_{1}(\gamma(t_{2}))\leq C_{\delta,\epsilon}\delta_{1}(\gamma(t_{1})).$

The proof of this Lemma 5.6 requires some lengthy averaging arguments and will
be deferred until Section 5.4, page 5.6. We stress that Lemma 5.6 is not
actually needed for any of the results in this work, and is proven only for
the sake of completeness of the literature review.

######  Remark 3.11.

The above formulation of Lemma 5.6 includes the Bianchi VIII case of
$N_{2},N_{3}>0>N_{1}$. The versions stated in [HU09b] and [Rin01] only
consider the Bianchi IX case $N_{1},N_{2},N_{3}>0$; however, their proofs
extend to this case virtually unchanged.

###### Proof of Theorem 1 using Lemma 5.6.

We begin by showing $\lim_{t\to\infty}\delta_{1}(t)=0$. By Lemma 3.9, we have
$\liminf_{t\to\infty}\delta_{1}(t)=0$. Suppose
$\limsup_{t\to\infty}\delta_{1}(t)>0$. Then $\delta_{1}$ must increase from
arbitrarily small values up to some finite nonzero infinitely often, and hence
we find arbitrarily late subintervals $[T^{L},T^{R}]\subseteq[0,\infty)$ such
that $\delta_{1}^{4}>|N_{1}N_{2}N_{3}|$ and $\delta_{1}$ increases by an
arbitrarily large factor. This contradicts Lemma 3.10 and Lemma 5.6, which
basically say that large increases of $\delta_{1}$ can neither happen with
$1+\Sigma_{+}>\epsilon_{+}$ nor with $1+\Sigma_{+}<\epsilon_{+}$.

The same applies for $\delta_{2}$ and $\delta_{3}$.

Suppose there was only a single $\omega$-limit point. This point cannot lie in
$\mathcal{K}\setminus\\{{\mathbf{T}}_{i}\\}$, since at each of these points,
at least one of the $N_{i}$ is unstable. The only remaining possibility is
$\omega({\bm{x}}_{0})={\mathbf{T}}_{i}$ for some $i\in\\{1,2,3\\}$, which is
excluded by Lemma 3.8. ∎

######  Remark 3.12.

The above proof also shows that $N_{2}N_{3}\to 0$ in the Bianchi VIII case
${\bm{x}}_{0}\in\mathcal{M}_{-++}\setminus\mathcal{T}_{1}$. This
generalization is directly possible while keeping [HU09b] virtually unchanged,
even though it has not been explicitly noted therein.

The above proof also shows that in Bianchi $\textsc{VII}_{0}$, i.e. for any
${\bm{x}}_{0}\in\mathcal{M}_{0++}$, we must have
$\lim_{t\to-\infty}{\bm{x}}(t)={\bm{p}}_{-}$ with ${\bm{p}}_{-}=(-1,0,0,N,N)$
for some $N>0$. This is false in the case of Bianchi $\textsc{VI}_{0}$: There
we have for any ${\bm{x}}_{0}\in\mathcal{M}_{0+-}$ that
$\lim_{t\to-\infty}{\bm{x}}(t)=(-1,0,0,0,0)$. Hence, $\delta_{1}$ can grow by
an arbitrarily large factor near ${\mathbf{T}}_{1}$ in $\mathcal{M}_{*+-}$,
and no analogue of Lemma 5.6 can hold in the Bianchi VIII models
$\mathcal{M}_{*+-}$ and $\mathcal{M}_{*-+}$.

This difficulty is partially responsible for the fact that, for
${\bm{x}}_{0}\in\mathcal{M}_{+-+}$, it was previously unknown whether
$\lim_{t\to\infty}N_{2}N_{3}(t)\overset{?}{=}0$ and
$\lim_{t\to\infty}N_{1}N_{3}(t)\overset{?}{=}0$.

##### Sketch of our replacement for Lemma 5.6.

In this work, we will replace the rather subtle averaging estimates from Lemma
5.6 by the program outlined in this paragraph. Let us first repeat the
reasons, why we want to avoid Lemma 5.6:

  1. 1.

The analogue statement of Lemma 5.6 in Bianchi VIII is wrong. Lemma 3.3 shows
that counterexamples to such a generalization can be found by taking any
sequence $\\{{\bm{x}}_{n}\\}$ of initial data converging to any point in
$\mathcal{M}_{0-+}$. Therefore, any argument relying on Lemma 5.6 has no
chance of carrying over to the Bianchi VIII case.

  2. 2.

The proof of Lemma 5.6 is lengthy and requires subtle averaging arguments.

  3. 3.

The complexity of the proof of Lemma 5.6 in not unavoidable: Most of the
effort is spent trying to understand asymptotic regimes _that do not occur
anyway_.

Our replacement is described by the following program:

  1. 1.

At first, we study pieces of trajectories
${\bm{x}}:[0,T]\to\mathcal{M}_{\pm\pm\pm}$, which start near $\mathcal{A}$ and
stay bounded away from the generalized Taub-spaces $\mathcal{T}^{G}_{i}$, i.e.
have all $r_{i}>\epsilon$. Along such partial solutions, all $\delta_{i}$
decay essentially exponentially (Proposition 4.1).

  2. 2.

Next, consider how solutions near $\mathcal{A}$ can enter the neighborhood of
the generalized Taub-spaces. This can only happen near some
$-{\mathbf{T}}_{i}$ (Proposition 4.2).

  3. 3.

For such solutions entering the vicinity of $-{\mathbf{T}}_{i}$, the quotient
$\frac{\delta_{i}}{r_{i}}$ is initially small, and stays small near
$-{\mathbf{T}}_{i}$ and along the heteroclinic leading to $+{\mathbf{T}}_{i}$
(Proposition 5.1).

  4. 4.

Next, we study solutions near ${\mathbf{T}}_{i}$ for which
$\frac{\delta_{i}}{r_{i}}$ is initially small. Then,
$\frac{\delta_{i}}{r_{i}}$ stays small. This additional condition
($\delta_{i}\ll r_{i}$) allows us to describe solutions with easier averaging
arguments and stronger conclusions than Lemma 5.6. Bianchi VIII solutions can
be analyzed same way. This is done in Section 5.3, leading to the conclusion
that $\delta_{i}$ decays essentially exponentially, with nonuniform rate
(Proposition 5.3).

  5. 5.

Finally, we combine the previous steps in Section 6 in order to prove Theorems
2, 3 and 4. These extend Theorem 1 with somewhat finer control over solutions
and provide an analogue in Bianchi VIII.

##  4 Dynamics near the Mixmaster-Attractor $\mathcal{A}$

Our previous arguments in Section 3 about the dynamics of (2.2.2) have been of
a rather qualitative and global character. We have established that there
exist $\omega$-limit points on the Mixmaster-attractor $\mathcal{A}$.

We have also sketched the classical proof that trajectories converge to
$\mathcal{A}$ in the case of Bianchi Type IX (Theorem 1) (where we deferred
the proof of the crucial estimate Lemma 5.6 to a later point).

In this section, we will give a more precise description of the behaviour near
$\mathcal{A}$. The goal of this section is to show that pieces of trajectories
$\gamma:[0,T]\to\mathcal{M}$ near $\mathcal{A}$ converge to $\mathcal{A}$
essentially exponentially, at least as long as they stay bounded away from the
Taub-points ${\mathbf{T}}_{i}$.

(a) The stability defining discs for the $N_{i}$

(b) The stability defining discs for the $\delta_{i}$

(c) A trajectory leaving the controlled regions

(d) The region Cap

(e) The region Circle

(f) The region Hyp

Figure 3: The relevant regions are colored in gray and are not up to scale
(except for Fig.2(a) and Fig.2(b)). See also Fig.2.

The goal of this section is to prove the following two propositions 4.1 and
4.2:

######  Proposition 4.1 (Essentially uniform exponential convergence to
$\mathcal{A}$ away from the Taub points).

For any $\varepsilon_{T}>0$ small enough, there exist constants
$\epsilon_{d},\epsilon_{N},\epsilon_{s},C_{0}>0$ (depending on
$\varepsilon_{T}$) and $C_{T}=5$ such that the following holds:

Consider a trajectory ${\bm{x}}:[0,T^{*})\to\mathcal{M}_{\pm\pm\pm}$, such
that, for all $t\in[0,T^{*})$ the following inequalities hold:
$\displaystyle\max_{i}\delta_{i}(t)$ $\displaystyle<\epsilon_{d}$ (4.1a)
$\displaystyle\min_{i}d({\bm{x}}(t),{\mathbf{T}}_{i})$
$\displaystyle>\varepsilon_{T}.$ (4.1b) Assume further that for the initial
condition ${\bm{x}}_{0}={\bm{x}}(0)$, the following stronger estimate holds:
$\max_{i}\delta_{i}({\bm{x}}_{0})<C_{0}^{-1}\epsilon_{d}$ (4.1c) Then, each
$\delta_{i}$ is essentially uniformly exponentially decreasing in $[0,T^{*})$,
i.e. $\delta_{i}(t_{2})\leq
C_{0}e^{-\epsilon_{s}(t_{2}-t_{1})}\delta_{i}(t_{1})\qquad\forall\,0\leq
t_{1}\leq t_{2}<T^{*},\quad i\in\\{1,2,3\\}.$ (4.2a) Hence, if $T^{*}<\infty$
and one of the inequalities (4.1b), (4.1a) is violated at time $T^{*}$, it
must be (4.1b), and (4.1a) must still hold at $T^{*}$.

Informally, this proposition states that trajectories near $\mathcal{A}$
converge exponentially to $\mathcal{A}$, as long as they stay bounded away
from the Taub points.

######  Proposition 4.2.

Assume the setting of Proposition 4.1. There are constants
$\epsilon_{N}=\epsilon_{N}(\varepsilon_{T})>0$ and $C_{T}=5$ such that
additionally the following holds:

Assume that $T^{*}<\infty$, and that initially
$\max_{i}d({\bm{x}}_{0},\mathcal{T}_{i})>C_{T}\varepsilon_{T}.$ (4.3a) Then
the final part of the trajectory preceding $T^{*}$ must have the form depicted
in Figure 3(c), i.e. there is $\ell\in\\{1,2,3\\}$ and there are times
$0<T_{1}<T_{2}<T_{3}\leq T^{*}$ (typically $T_{3}=T^{*}$) such that
$\displaystyle d({\bm{x}}(t),-{\mathbf{T}}_{\ell})$ $\displaystyle\leq
C_{T}\varepsilon_{T}$ $\displaystyle\forall t\in[T_{1},T_{2}]$ (4.4a)
$\displaystyle\max_{i}|N_{i}(t)|$ $\displaystyle\geq\epsilon_{N}$
$\displaystyle\forall t\in[T_{2},T_{3}]$ (4.4b) $\displaystyle
d({\bm{x}}(t),{\mathbf{T}}_{\ell})$ $\displaystyle\leq C_{T}\varepsilon_{T}$
$\displaystyle\forall t\in[T_{3},T^{*}].$ (4.4c)

Informally, this proposition states that the only way for trajectories near
$\mathcal{A}$ to reach the vicinity of the Taub-points is via the heteroclinic
connection $-{\mathbf{T}}_{i}\to{\mathbf{T}}_{i}$.

We first give an informal outline of the proofs:

###### Informal proof of Proposition 4.1.

We split the trajectory into time intervals where it is either near
$\mathcal{K}$ (i.e. $\max_{i}|N_{i}|\leq\epsilon_{N}$) or away from
$\mathcal{K}$ (i.e. $\max_{i}|N_{i}|\geq\epsilon_{N}$).

Near $\mathcal{K}$, we can see from (2.2.2b) and (2.2.8) that each
${\mathrm{D}}_{t}\log|N_{i}|$ and ${\mathrm{D}}_{t}\log\delta_{i}$ depends
only on the ${\bm{\Sigma}}$-coordinates and is positive only on some disc in
${\mathbb{R}}^{2}$. These six discs are plotted in Figure 2(a). We can observe
that these discs only touch and intersect $\mathcal{K}$ at the three Taub-
points and that near each point
${\bm{p}}\in\mathcal{K}\setminus\\{{\mathbf{T}}_{i}\\}$, exactly one of the
$\log|N_{i}|$ is increasing and the two remaining $\log|N_{j}|$ and all three
$\log\delta_{i}$ are decreasing. Under our assumption
$\min_{i}d(\gamma(t),{\mathbf{T}}_{i})>\varepsilon_{T}$, this increase and
decrease is uniform when $\max_{i}|N_{i}|\leq\epsilon_{N}$ and if
$\epsilon_{N}=\epsilon_{N}(\varepsilon_{\mathbf{T}})>0$ is small enough.
Hence, for any small piece of trajectory
${\bm{x}}:[t_{1},t_{2}]\to\\{{\bm{x}}\in\mathcal{M}:\,\min_{i}d({\bm{x}},{\mathbf{T}}_{i})\geq\varepsilon_{T},\,\max_{i}|N_{i}|\leq\epsilon_{N}\\}$,
one of the $|N_{i}|$ is uniformly exponentially increasing, while all three
$\delta_{i}$ and the remaining two $|N_{j}|,|N_{k}|$ are uniformly
exponentially decreasing, with some rate
$2\epsilon_{s}=2\epsilon_{s}(\varepsilon_{T},\epsilon_{N})>0$.

Eventually the trajectory will leave the neighborhood of $\mathcal{K}$; since
we assumed that we are near $\mathcal{A}$, i.e.
$\max_{i}\delta_{i}<\epsilon_{d}$, this can only happen near one of the Kasner
caps (Bianchi type II). By continuity of the flow, the trajectory will follow
a heteroclinic orbit until it is near $\mathcal{K}$ again, and will spend only
bounded amount of time $T<C(\epsilon_{N})$ for this transit. Hence, all
$|N_{i}|$ and $\delta_{i}$ can only change by a bounded factor during such a
heteroclinic transit.

The time spent near $\mathcal{K}$ between two heteroclinic transits is bounded
below by $\mathcal{O}(|\log\epsilon_{d}|)$ (for fixed $\epsilon_{N}$):
Consider an interval $[t_{1},t_{2}]$ spent near $\mathcal{K}$, where
$t_{1}>0$. Suppose without loss of generality that initially
$|N_{1}|(t_{1})=\epsilon_{N}$ and that $|N_{2}|$ is uniformly exponentially
increasing, such that $|N_{2}|(t_{2})=\epsilon_{N}$. Then, we must have
$|N_{2}|(t_{1})=\frac{4|N_{1}|}{\delta_{3}^{2}}(t_{1})\leq
C\epsilon_{N}\epsilon_{d}^{-2}$, and we must have $t_{2}-t_{1}\geq
C|\log\epsilon_{d}|$. Hence, if $\epsilon_{d}\ll\epsilon_{N}$ is small enough,
the exponential decrease of the three $\delta_{i}$ will dominate all
contributions from the heteroclinic transits and we obtain an estimate of the
form (4.2a). ∎

###### Informal proof of Proposition 4.2.

We again use continuity of the flow: Each small heteroclinic “bounce” near one
of the ${\mathbf{T}}_{i}$ must increase the distance from ${\mathbf{T}}_{i}$
by at least some $C_{u}=C_{u}(\varepsilon_{T})$. By continuity of the flow,
each episode with $\max_{i}|N_{i}|\geq\epsilon_{N}$ therefore must increase
the distance from ${\mathbf{T}}_{i}$ by $C_{u}/2$; near $\mathcal{K}$, the
trajectory is almost constant and $d({\bm{x}}_{0},{\mathbf{T}}_{i})$ cannot
shrink by more that $C_{u}/3$. Hence the only way to reach the vicinity of a
Taub point is by following the heteroclinic
$-{\mathbf{T}}_{i}\to{\mathbf{T}}_{i}$. ∎

The remainder of this section is devoted to making these informal proofs
rigorous, i.e. filling all the gaps and replacing the hand-wavy arguments by
formal ones. We begin by naming the regions of the phase-space, where the
various estimates hold:

######  Definition 4.3.

Given $\varepsilon_{\mathbf{T}},\epsilon_{N},\epsilon_{d}>0$ (later chosen in
this order) we define:

$\displaystyle{\textsc{Cap}}[\epsilon_{N},\epsilon_{d}]$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:\max|N_{i}|\geq\epsilon_{N},\max_{i}\delta_{i}\leq\epsilon_{d}\\}$
(4.5) $\displaystyle{\textsc{Circle}}[\epsilon_{N},\epsilon_{d}]$
$\displaystyle=\\{{\bm{x}}\in\mathcal{M}:\max|N_{i}|\leq\epsilon_{N},\,\max_{i}\delta_{i}\leq\epsilon_{d}\\}$
$\displaystyle{\textsc{Hyp}}[\varepsilon_{\mathbf{T}},\epsilon_{N},\epsilon_{d}]$
$\displaystyle={\textsc{Circle}}[\epsilon_{N},\epsilon_{d}]\setminus\left[B_{\varepsilon_{\mathbf{T}}}({\mathbf{T}}_{1})\cup
B_{\varepsilon_{\mathbf{T}}}({\mathbf{T}}_{2})\cup
B_{\varepsilon_{\mathbf{T}}}({\mathbf{T}}_{3})\right].$

These sets are sketched in Figure 3 (not up to scale). They are constructed
such that for appropriate parameter choices:

  1. 1.

The union $\textsc{Cap}\cup\textsc{Circle}$ contains an entire neighborhood of
$\mathcal{A}$ (by construction).

  2. 2.

The region Circle is a small neighborhood of the Kasner circle. This is
because of the constraint $1=\Sigma^{2}+N^{2}$ and $|N^{2}|<C\epsilon_{N}^{2}$
(see Figure 3(e)).

  3. 3.

The region Cap has three connected components, where one of the three
$|N_{i}|\gg 0$, because by $\max_{i}|N_{i}|\geq\epsilon_{N}$ at least one
$N_{i}$ must be bounded away from zero and by
$\max_{i}2\sqrt{|N_{j}N_{k}|}\leq\epsilon_{d}$ at most one $N_{i}$ can be
bounded away from zero (this only works if $\epsilon_{d}$ is small enough,
depending on $\epsilon_{N}$).

The region Cap is bounded away from the Kasner circle (see Figure 3(d)). By
continuity of the flow, the dynamics in Cap can be approximated by pieces of
heteroclinic orbits in $\mathcal{A}$, up to uniformly small errors (Lemma
4.5).

  4. 4.

The region Hyp has three connected components. In each connected component,
one of the $|N_{i}|$ is uniformly exponentially increasing and the remaining
two $|N_{j}|$, $|N_{k}|$ are uniformly exponentially decreasing. All three
products $\delta_{i}$ are uniformly exponentially decreasing in Hyp (Lemma
4.4; this only works if $\epsilon_{N}$ is small enough, depending on
$\varepsilon_{\mathbf{T}}$).

  5. 5.

The remaining part of the neighborhood of $\mathcal{A}$, i.e.
$\textsc{Circle}\setminus\textsc{Hyp}$, consists of the neighborhoods of the
three Taub points. The analysis of the dynamics in these neighborhoods is
deferred until Section 5.

######  Lemma 4.4 (Uniform Hyperbolicity Estimates).

Given any $\varepsilon_{T}>0$ small enough, we find $\epsilon_{N}>0$ and
$\epsilon_{s}>0$ small enough such that, for any
${\bm{x}}\in\textsc{Hyp}[\varepsilon_{{\mathbf{T}}},\epsilon_{N},\infty]$, we
find one $i\in\\{1,2,3\\}$ such that
${\mathrm{D}}_{t}\log|N_{i}|>2\epsilon_{s}$, and the remaining two
${\mathrm{D}}_{t}\log|N_{j}|<-2\epsilon_{s}$ and all three
${\mathrm{D}}_{t}\log\delta_{j}<-2\epsilon_{s}$.

Let $\varepsilon_{\mathbf{T}},\epsilon_{N},\epsilon_{s}>0$ as above. For any
piece of trajectory
${\bm{x}}:(t_{1},t_{2})\to\textsc{Hyp}[\epsilon_{N},\infty]$, we can conclude

$\displaystyle\int_{t_{1}}^{t_{2}}|N_{i}|{\mathrm{d}}t$
$\displaystyle<\frac{\epsilon_{N}}{2\epsilon_{s}}$ (4.6)
$\displaystyle\mathrm{diam}\,\gamma\leq\int_{t_{1}}^{t_{2}}|\gamma^{\prime}(t)|{\mathrm{d}}t$
$\displaystyle\leq
C_{\Sigma}\int_{t_{1}}^{t_{2}}\max_{i}|N_{i}|(t){\mathrm{d}}t\leq
C_{\Sigma}\frac{\epsilon_{N}}{2\epsilon_{s}},$

where we can choose $C_{\Sigma}=2$ if $\epsilon_{N}<0.1$ (from (2.2.2)).

###### Proof.

The first part of the lemma consists of choosing $\epsilon_{s}$ and
$\epsilon_{N}$ dependent on $\varepsilon_{T}$. From Equations (2.2.2b) and
(2.2.8), we see that each ${\mathrm{D}}_{t}\log|N_{i}|$ and
${\mathrm{D}}_{t}\log\delta_{i}$ depends only on the
${\bm{\Sigma}}$-coordinates and is positive only on some disc in
${\mathbb{R}}^{2}$. These six discs are plotted in Figure 2(a), and only touch
or intersect $\mathcal{K}$ near the three Taub-points, a neighborhood of which
is excluded. By the constraint $1-\Sigma^{2}=N^{2}$ and $|N^{2}|\leq
9\epsilon_{N}^{2}$, the set Hyp, depicted in Figure 3, is near $\mathcal{K}$,
and the desired uniformity estimates hold.

The second part follows from the uniform hyperbolicity in Hyp: In each
component of Hyp, exactly one $N_{i}$ is unstable (see Figure 3 and Figure
2(a)). Suppose without loss of generality that $N_{1}$ is the unstable
direction; then we can estimate for $t\in(t_{1},t_{2})$:

$|N_{1}(t)|\leq e^{-2\epsilon_{s}(t_{2}-t)}|N_{1}(t_{2})|,\qquad|N_{2}(t)|\leq
e^{-2\epsilon_{s}(t-t_{1})}|N_{2}(t_{1})|.$

For $|N_{3}|$, the analogous estimate as for $N_{2}$ holds. Using
$|N_{1}(t_{2})|\leq\epsilon_{N}$ and $|N_{2}(t_{1})|\leq\epsilon_{N}$ and
integrating yields the claim about $\int|N|{\mathrm{d}}t$. From (2.2.2), we
see $|{\bm{x}}^{\prime}|\leq C_{\Sigma}\max_{i}|N_{i}|$ (if
$\max_{i}|N_{i}|\leq 1$). ∎

Continuity of the flow allows us to approximate solutions in Cap by
heteroclinic solutions in $\mathcal{A}$, up to any desired precision
$\varepsilon_{c}$, if we only chose the distance from $\mathcal{A}$ (i.e.
$\epsilon_{d}>0$) small enough. More precisely:

######  Lemma 4.5 (Continuity of the flow near Cap).

Let $\epsilon_{N}>0$ and $\varepsilon_{c}>0$. Then there exists
$\epsilon_{d}=\epsilon_{d}(\epsilon_{N},\varepsilon_{c})$ small enough and
$\hat{C}_{0}=\hat{C}_{0}(\epsilon_{N})>0$ large enough, such that the
following holds:

Let ${\bm{x}}:(t_{1},t_{2})\to\textsc{Cap}[\epsilon_{N},\epsilon_{d}]$ be a
piece of a trajectory. Then $t_{2}-t_{1}<\hat{C}_{0}$ and there exists
$y\in\mathcal{A}$ such that

$d({\bm{x}}(t),\phi(y,t-t_{1}))<\varepsilon_{c}\qquad\text{for all}\quad
t\in(t_{1},t_{2}),$ (4.7)

where $\phi:\mathcal{M}\times{\mathbb{R}}\to\mathcal{M}$ is the flow
corresponding to (2.2.2).

###### Proof.

Follows from continuity of the flow and the fact that all trajectories in
$\mathcal{A}$ are heteroclinic and must leave Hyp at some time. ∎

We now have collected all the ingredients to formally prove the two main
results from this section. At first, we combine Lemma 4.4 and Lemma 4.5 in
order to show that each $\delta_{i}$ is uniformly essentially exponentially
decreasing in $\textsc{Cap}\cup\textsc{Hyp}$:

###### Formal proof of Proposition 4.1.

Given $\varepsilon_{T}>0$, find
$\epsilon_{N},\epsilon_{d},\epsilon_{s},\hat{C}_{0}>0$ such that Lemma 4.4 and
Lemma 4.5 hold (for arbitrary $\varepsilon_{c}$).

Set $\mu_{i}=\delta_{i}^{\prime}/\delta_{i}+\epsilon_{s}$; it suffices to show
that $\int_{t_{1}}^{t_{2}}\mu(t){\mathrm{d}}t<\log C_{0}$ is bounded above,
independently of $\gamma$, $i\in\\{1,2,3\\}$ and $t_{1}$,$t_{2}$. Fix $\gamma$
and $t_{1}<t_{2}$.

Decompose $[t_{1},t_{2}]$ into intervals $S_{k}<T_{k}<S_{k+1}$ corresponding
to the preimages of the regions Cap and Hyp, i.e. such that

${\bm{x}}([S_{k},T_{k}])\subseteq{\textsc{Hyp}}\qquad\text{and}\qquad{\bm{x}}([T_{k},S_{k+1}])\subseteq{\textsc{Cap}}.$

We begin by considering the contribution from Cap, i.e., an interval
$[T_{k},S_{k+1}]$. By Lemma 4.5 we have $S_{k+1}-T_{k}<\hat{C}_{0}$; since
$\mu$ is bounded, we get $\int_{T_{k}}^{S_{k+1}}\mu(t){\mathrm{d}}t<\log
C_{0}$ for some $C_{0}$.

Next, we consider the contributions from Hyp. In this region,
$\mu<-\epsilon_{s}$. Take an interval $[S_{k},T_{k}]$, which is not the
initial or final interval, i.e. $t_{1}<S_{k}<T_{k}<t_{2}$. Assume without loss
of generality that $|N_{1}(S_{k})|=\epsilon_{N}$ and
$|N_{2}(T_{k})|=\epsilon_{N}$. Then
$|N_{2}(S_{k})|=0.25\delta_{3}^{2}/|N_{1}|(S_{k})<\epsilon_{d}^{2}/\epsilon_{N}$.
Since $|N_{2}^{\prime}/N_{2}|<3$, we obtain
$T_{k}-S_{k}>-\frac{2}{3}\log\frac{\epsilon_{d}}{\epsilon_{N}}$. Adjust
$\epsilon_{d}>0$ to be so small, that $C_{s}(T_{k}-S_{k})>2\log C_{0}$. Then
such an interval gives us a contribution of
$\int_{S_{k}}^{T_{k}}\mu(t){\mathrm{d}}t<-\log C_{0}$.

For the complete interval $(t_{1},t_{2})$, sum over $k$; two disjoint
intervals in the Cap-region must always enclose an interval in the Hyp-
section, which cancels the contribution of its preceding Cap-region.
Therefore, at most the last Cap-region stays unmatched and we obtain
$\int_{t_{1}}^{t_{2}}\mu(t){\mathrm{d}}t<\log C_{0}.$ ∎

Next, we adjust the constants from Lemma 4.5 in order to show that
trajectories near $\mathcal{A}$ can only enter the vicinity of Taub-points via
the heteroclinic $-{\mathbf{T}}_{\ell}\to{\mathbf{T}}_{\ell}$:

###### Formal proof of Proposition 4.2.

We find some $C_{u}>0$ such that $d(K(p),T_{i})>d(p,T_{i})+C_{u}$ for every
$p\in\mathcal{K}$ with $d(p,{\mathbf{T}}_{i})\in(\varepsilon_{T},0.5]$. It is
evident from Figure 2(a) (or, formally, Proposition 3.2) that this is
possible.

By Lemma 4.4, we can make $\epsilon_{N}$ small enough that
$\mathrm{diam}\,\gamma<C_{u}/8$ for pieces of trajectories
$\gamma:(t_{1},t_{2})\to{\textsc{Hyp}}$. Using the continuity of the flow,
i.e. Lemma 4.5, we can make $\epsilon_{d}$ small enough such that pieces
${\bm{x}}:(t_{1},t_{2})\to{\textsc{Cap}}$ are approximated by heteroclinic
orbits up to distance $C_{u}/4$.

Now suppose $T^{*}<\infty$ and
$\min_{i}d({\bm{x}}_{0},\mathcal{T}_{i})>5\varepsilon_{T}$. We cannot have
$\max_{i}\delta_{i}(\phi(x_{0},T^{*}))\geq\epsilon_{d}$; hence,
$d({\bm{x}}(T^{*}),{\mathbf{T}}_{\ell})=\varepsilon_{T}$ for some
$\ell\in\\{1,2,3\\}$. Set

$T^{3}=\sup\left\\{t<T^{*}:{\bm{x}}(t)\not\in\overline{B_{1.5\varepsilon_{T}}({\mathbf{T}}_{\ell})}\cap\textsc{Circle}[\epsilon_{n},\epsilon_{d}]\right\\}.$

By the assumption (4.3a), we have $T^{3}>0$. We cannot have
$d({\mathbf{T}}_{\ell},{\bm{x}}(T^{3}))=1.5\varepsilon_{\mathbf{T}}$, since we
already know $\mathrm{diam}\,{\bm{x}}([T^{3},T^{*}])\leq
C_{u}/8<0.5\varepsilon_{T}$ (since, by construction,
${\bm{x}}([T^{3},T^{*}])\subseteq\textsc{Hyp}(\varepsilon_{\mathbf{T}},\epsilon_{N},\epsilon_{d})$).
This proves (4.4c), as well as

${\bm{x}}(T^{3})\in\partial\textsc{Cap}[\epsilon_{N},\epsilon_{d}]\cap\partial\textsc{Circle}[\epsilon_{N},\epsilon_{d}]\cap
B_{1.5\varepsilon_{T}}({\mathbf{T}}_{\ell}).$

Next, we set

$T^{2}=\sup\left\\{t\in[0,T^{3}):{\bm{x}}([t,T_{3}))\subseteq\textsc{Cap}[\epsilon_{n},\epsilon_{d}]\right\\}.$

In the interval $t\in[T_{2},T_{3}]$, the trajectory is in one of the three Cap
regions; this must be the $|N_{\ell}|\geq\epsilon_{N}$ cap, since otherwise
$d({\bm{x}}(t),{\mathbf{T}}_{\ell})$ would be decreasing (see Figure 2(a)). We
set

$T^{1}=\sup\left\\{t\in[0,T^{2}):{\bm{x}}(t)\not\in\textsc{Circle}[\epsilon_{n},\epsilon_{d}]\right\\}.$

Similar arguments yield the remaining claim (4.4a). ∎

######  Remark 4.6.

The constants generated in this section are sub-optimal (at least doubly
exponentially so). If one cared at all about their numerical values, then one
would need to replace Lemma 4.5 and Proposition 3.2 by explicit estimates.

##  5 Analysis near the generalized Taub-spaces $\mathcal{T}_{i}^{G}$

In this section, we will study the dynamics in the vicinity of the generalized
Taub-spaces, without loss of generality $\mathcal{T}_{1}^{G}$, using the polar
coordinates from Section 2.3. This section is structured in the following way:

In section 5.1, we will give a highly informal motivation for the general form
of our estimates. This part can be safely skipped by readers who are
uncomfortable with its hand-wavy nature. In section 5.2, we will study the
behaviour of trajectories near the heteroclinic orbit
$-{\mathbf{T}}_{1}\to{\mathbf{T}}_{1}$, which come from either the $|N_{2}|\gg
0$ or the $|N_{3}|\gg 0$ cap. In section 5.3, we will study the further
behaviour near ${\mathbf{T}}_{1}$ of such trajectories. In section 5.4, we
will study the behaviour of trajectories near ${\mathbf{T}}_{1}$ which do
_not_ necessarily have the prehistory described in section 5.2, and especially
provide the deferred proof of Lemma 5.6. This section is mostly optional for
our main results: Any trajectory which ever leaves the region where Section
5.4 is necessary will never revisit this region, a fact which is proven
without referring to any results from Section 5.4.

###  5.1 Informal Motivation

We already alluded to the motivation for the estimates in this section in the
introductory Section 1 , page 1: From Proposition 4.1, by varying
$\varepsilon_{\mathbf{T}}$, we can control the behaviour of trajectories near
${\mathbf{T}}_{1}$ if $\delta_{1}\ll r_{1}$ and obtain estimates of the
following type for partial trajectories $\gamma:(t_{1},t_{2})\to
B_{\epsilon}({\mathbf{T}}_{1})$ and continuous monotonous functions
$\rho:(0,1]\to(0,1]$:

> Suppose $\delta_{1}(t_{1})<\rho_{0}(r_{1}(t_{1}))$;
> then $\delta_{1}(t_{2})<C_{0}e^{-\rho_{1}(r_{1}(t_{1}))}\delta_{1}(t_{1})$
> and $r_{1}(t_{2})\geq\rho_{2}(r_{1}(t_{1}))$.

The bounds will take the specific form $\rho_{0}(r)=Cr$ and $\rho_{2}(r)=Cr$
and $\rho_{1}(r)=\frac{C}{r^{2}}$ (Proposition 5.3).

We know some prehistory of trajectories entering the vicinity of
${\mathbf{T}}_{1}$ (by Proposition 4.2), which allows us to track backwards
the condition $\delta_{1}<\rho_{0}(r_{1})$. Further tracking back this
condition, it is clear that trajectories entering the vicinity of
$-{\mathbf{T}}_{1}$ must have $\delta_{1}\leq\epsilon_{d}\ll
r_{1}\sim\epsilon_{N}$.

At least in the Bianchi IX-like case, where
$\mathrm{sign}\,N_{2}=\mathrm{sign}\,N_{3}$, the set
$\\{{\bm{x}}:\,r_{1}=0\\}$ is invariant, allowing us to get some
$\rho_{3}:(0,1]\to(0,1]$ such that $\delta_{1}<\rho_{3}(r_{1})$ for _any_
trajectory entering the vicinity of ${\mathbf{T}}_{1}$ via the route in
Proposition 4.2, i.e. via $-{\mathbf{T}}_{1}$ and some cap before.

These estimates combine well if we can make $\rho_{3}<\rho_{0}$. The estimates
will take the specific form $\rho_{3}(r)=\epsilon r$, for arbitrarily small
$\epsilon>0$ (Proposition 5.1), which is precisely the required estimate at
${\mathbf{T}}_{1}$. In Bianchi VIII, we have no qualitative a-priori reason to
expect bounds of the same form. Nevertheless, we will prove that they hold,
which allows us to control any solution entering ${\mathbf{T}}_{1}$ as in
Proposition 4.2.

For the sake of brevity of arguments, we will present our analysis in the
reverse order: We chronologically follow a trajectory from $-{\mathbf{T}}_{1}$
to $+{\mathbf{T}}_{1}$ and then until it leaves the vicinity of
$+{\mathbf{T}}_{1}$, instead of tracking estimates backwards.

###  5.2 Analysis near $-{\mathbf{T}}_{1}$ and near the heteroclinic
$-{\mathbf{T}}_{1}\to{\mathbf{T}}_{1}$

The behaviour of trajectories away from ${\mathbf{T}}_{1}$ is already
partially described by Proposition 4.1; we only need to additionally estimate
the quotient $\frac{\delta_{1}}{r_{1}}$ in this region. The necessary
estimates can be summarized in the following:

######  Proposition 5.1.

Let $\varepsilon_{T}\in(0,0.1)$. We can chose $\epsilon_{N},\epsilon_{d}>0$
small enough, such that Propositions 4.1 and 4.2 hold, as well as choose
constants $C_{1},\ldots,C_{5}>0$ large enough, such that the following holds:

Let $0<T_{1}\leq T_{2}$ and ${\bm{x}}:[0,T_{2})\to M_{\pm,\pm,\pm}$ be piece
of trajectory, such that

$\displaystyle{\bm{x}}(0)$
$\displaystyle\in\partial\textsc{Circle}[\epsilon_{N},\epsilon_{d}],$ (5.2.1a)
$\displaystyle\quad\text{i.e.}\quad\max_{i}\delta_{i}(0)<\epsilon_{d},\
\max(|N_{2}|,|N_{3}|)(0)=\epsilon_{N}$ $\displaystyle{\bm{x}}([0,T_{1}))$
$\displaystyle\subseteq
B_{2\varepsilon_{T}}(-{\mathbf{T}}_{1})\cap\textsc{Circle}[\epsilon_{N},\epsilon_{d}],$
(5.2.1b)
$\displaystyle\quad\text{i.e.}\quad\max_{i}\delta_{i}(t)<\epsilon_{d},\
\max_{i}|N_{i}|(t)\leq\epsilon_{N},\ d(\gamma(t),{\mathbf{T}}_{1})\leq
2\varepsilon_{T}\qquad\forall\,t\in[0,T_{1}]$
$\displaystyle{\bm{x}}([T_{1},T_{2}))$
$\displaystyle\subseteq\textsc{Cap}[\epsilon_{N},\epsilon_{d}],$ (5.2.1c)
$\displaystyle\quad\text{i.e.}\quad\max_{i}\delta_{i}(t)<\epsilon_{d},\
|N_{1}|(t)\geq\epsilon_{N}\quad\forall\,t\in[T_{1},T_{2}),$

i.e. we are in the situation of the conclusion of Proposition 4.2.

Then the following estimates hold:

$\displaystyle\frac{\delta_{1}}{r_{1}}(0)$ $\displaystyle\leq
C_{1}\epsilon_{d}$ (5.2.2a) $\displaystyle\frac{\delta_{1}}{r_{1}}(t_{2})$
$\displaystyle\leq C_{2}\frac{\delta_{1}}{r_{1}}(t_{1})\qquad$
$\displaystyle\forall$ $\displaystyle 0\leq t_{1}\leq t_{2}<T_{2}$ (5.2.2b)
$\displaystyle\delta_{1}(t_{2})$ $\displaystyle\leq
C_{3}e^{-C_{4}^{-1}(t_{2}-t_{1})}\delta_{1}(t_{1})\qquad$
$\displaystyle\forall$ $\displaystyle 0\leq t_{1}\leq t_{2}<T_{2}$ (5.2.2c)
$\displaystyle r_{1}(t_{2})$ $\displaystyle\geq C_{5}^{-1}r_{1}(t_{1})\qquad$
$\displaystyle\forall$ $\displaystyle T_{1}\leq t_{1}\leq t_{2}<T_{2}.$
(5.2.2d)

Alternatively to the assumption (5.2.1a), we can assume (5.2.2a); then the
case $0=T_{1}$ is valid as well.

###### Proof.

The claim (5.2.2c) is already proven in Proposition 4.1. Assume without loss
of generality that $|N_{3}|(0)=\epsilon_{N}$. Assuming that $\epsilon_{d}$ is
small enough compared to $\epsilon_{N}$, we have
$|N_{2}|(0)\leq\delta_{1}^{2}(0)/\epsilon_{N}\leq 0.5\epsilon_{N}$ and hence
$r_{1}(0)\geq|N_{3}|-|N_{2}|\geq 0.5\epsilon_{N}$. Therefore
$\frac{\delta_{1}}{r_{1}}(0)\leq C\delta_{1}(0)\leq C\epsilon_{d}$ and claim
(5.2.2a) holds. First, consider the Bianchi IX-like case
${{\hat{n}}}_{2}={{\hat{n}}}_{3}$ in polar coordinates, i.e. equations
(2.3.1). We can immediately estimate
$\partial_{t}\log\frac{\delta_{1}}{r_{1}}\leq C|N_{1}|$. We already
established in Section 4 that $\int|N_{1}|{\mathrm{d}}t$ is bounded for
$t_{1},t_{2}\in[0,T_{1}]$ (Lemma 4.4) and that $T_{2}-T_{1}$ is bounded (Lemma
4.5), yielding (5.2.2b).

Next, consider the Bianchi VIII-like case ${{\hat{n}}}_{2}=+1$ and
${{\hat{n}}}_{3}=-1$, i.e. equations (2.3.4d). Assume that we have for all
$t\in[0,T_{2}]$,

$\frac{\delta_{1}}{r_{1}}(t)<1.$ (5.2.3)

Under the assumption (5.2.3), we can estimate

$\frac{N_{+}}{r_{1}}=\frac{\sqrt{N_{-}^{2}+\delta_{1}^{2}}}{r_{1}}=\sqrt{\frac{N_{-}^{2}}{N_{-}^{2}+\Sigma_{-}^{2}}+\frac{\delta_{1}^{2}}{r_{1}^{2}}}\leq\sqrt{2},$

and hence $\partial_{t}\log\frac{\delta_{1}}{r_{1}}\leq C|N_{1}|$, yielding
(5.2.2b). When we adjust $\varepsilon_{d}$ such that
$C_{1}C_{2}\varepsilon_{d}<1$, then this argument bootstraps to prove
(5.2.2b), without assuming a priori (5.2.3) (proof: Assume there was a time
$T\in(0,T_{2})$ such that (5.2.3) was violated; then
$\frac{\delta_{1}}{r_{1}}(T)\leq C_{2}\frac{\delta_{1}}{r_{1}}(0)\leq
C_{2}C_{1}\varepsilon_{d}<1$).

Next, we prove the remaining claim (5.2.2d). Since we know that
$\frac{\delta_{1}}{r_{1}}(t)<1$, we can estimate ${\mathrm{D}}_{t}\log r<C$
for for all $t\in[T_{1},T_{2})$, both in Bianchi VIII and IX. Since
$T_{2}-T_{1}$ is bounded, we obtain (5.2.2d).

Considering the above proof, it is obvious that we can alternatively replace
the assumption (5.2.1a) by (5.2.2a), and then also allow $0=T_{1}$. ∎

######  Remark 5.2.

We excluded the set $\textsc{Inaccessible}[\epsilon_{v}]$ from our analysis,
given by

$\textsc{Inaccessible}[\epsilon_{v}]=\\{{\bm{x}}\in{\mathbb{R}}^{5}:\delta_{1}\geq\epsilon_{v}r_{1}\\},$

i.e. we described the dynamics outside of Inaccessible and showed that the set
Inaccessible cannot reached by initial conditions described by Proposition
4.2.

Ignoring the constraint $G=1$, the set Inaccessible looks like a linear cone
times ${\mathbb{R}}^{2}$, since both $r_{1}$ and $\delta_{1}$ are homogeneous
of first order in $N_{2},N_{3},\Sigma_{-}$ and independent of $N_{1}$ and
$\Sigma_{+}$.

Even though Bianchi VIII lacks an explicit invariant Taub-space, the
Inaccessible-cone around the generalized Taub-space $\mathcal{T}_{1}^{G}$ is a
suitable “morally backwards invariant” replacement.

###  5.3 Analysis near ${\mathbf{T}}_{1}$

Our analysis of the neighborhood of ${\mathbf{T}}_{1}$ can be summarized in
the following

######  Proposition 5.3.

For any $\varepsilon_{T}\in(0,0.1)$, there exist constants $\epsilon_{v}>0$,
$C_{1},C_{r},C_{\delta,r}>0$ and $C_{e}=10$ such that the following holds:

Let $\gamma:[0,T^{*})\to B_{2\varepsilon_{T}}({\mathbf{T}}_{1})$ be a partial
trajectory with

$\frac{\delta_{1}}{r_{1}}(0)<\epsilon_{v}.$ (5.3.1)

Then, for all $0\leq t_{1}\leq t_{2}<T^{*}$:

$\displaystyle(|N_{1}|,\delta_{2},\delta_{3})(t_{2})$ $\displaystyle\leq
C_{1}e^{-C_{e}^{-1}(t_{2}-t_{1})}\,(|N_{1}|,\delta_{2},\delta_{3})(t_{1})$
(5.3.2a) $\displaystyle\frac{\delta_{1}}{r_{1}}(t_{2})$ $\displaystyle\leq
C_{\delta,r}\frac{\delta_{1}}{r_{1}}(t_{1})$ (5.3.2b) $\displaystyle
r_{1}(t_{2})$ $\displaystyle\geq C_{r}^{-1}r_{1}(t_{1})$ (5.3.2c)
$\displaystyle\delta_{1}(t_{2})$ $\displaystyle\leq
C_{0}\exp\left(-\frac{C_{e}^{-1}}{r_{1}^{2}(t_{1})}(t_{2}-t_{1})\right)\,\delta_{1}(t_{1})$
(5.3.2d) $\displaystyle T^{*}$ $\displaystyle<\infty.$ (5.3.2e)

We begin by proving the first three of the claims, in a way analogous to the
proof of Proposition 5.1:

###### Proof of Proposition 5.3, conclusions (5.3.2a), (5.3.2b), (5.3.2c).

The exponential decay (5.3.2a) follows trivially from (2.2.2b), see e.g.
Figures 3(a) and 2(a).

In order to see (5.3.2b), we again need to bootstrap: First consider
$T_{1}=\sup\\{t\in[0,T^{*}):\frac{\delta_{1}}{r_{1}}(t^{\prime})<1\,\forall
t^{\prime}\in[0,t)\\}$. We can estimate
$\partial_{t}\log\frac{\delta_{1}}{r_{1}}<C|N_{1}|$ for all $t\in[0,T_{1})$,
using (2.3.2c) and (2.3.5c) and the fact that the higher order terms
$|h_{r}|,|h_{\delta}|,|h_{\psi}|$ are bounded (by $C=5$). By the exponential
decay of $N_{1}$, this yields (5.3.2b) upon integration, for all $t_{2}\leq
T_{1}$. By adjusting $\epsilon_{v}<C_{\delta,r}^{-1}$ we can then conclude
$T_{1}=T^{*}$.

Using $\frac{\delta_{1}}{r_{1}}<1$, we can estimate
$\partial_{t}r_{1}>-C|N_{1}|$, which upon integration yields the claim
(5.3.2c). ∎

The next estimate (5.3.2d) requires a slightly more involved averaging-style
argument, similar to the proof of Proposition 4.1:

###### Proof of Theorem 5.3, conclusion (5.3.2d).

We set

$\mu={\mathrm{D}}_{t}\log\delta_{1}+0.1r_{1}^{2}.$

It suffices to prove that $\int_{t_{1}}^{t_{2}}\mu(t){\mathrm{d}}t\leq\log
C_{0}$ for some $C_{0}>0$.

##### Strategy.

We will first consider times where $|N_{1}|\not\ll r_{1}$; the integral
$\int\mu{\mathrm{d}}t$ over these times will be bounded by
$\int|N_{1}|{\mathrm{d}}t$. Next we will split $\mu$ into a nonpositive and a
nonnegative part; the nonnegative (bad) part will have a contribution for
every $\psi$ rotation, which is bounded by $Cr_{1}$, while the nonpositive
(good) part will have a negative contribution for every $\psi$-rotation which
scales with $r_{1}\log\frac{\delta_{1}}{r_{1}}$. Adjusting $\epsilon_{v}$ will
then yield the desired estimate (after summing over $\psi$-rotations).

##### Estimates for large $|N_{1}|$.

Choose $\widetilde{T}$ (possibly $\widetilde{T}=0$) such that $|N_{1}(t)|\geq
Cr_{1}^{2}(t)$ (with $C=0.05$) for $t\in(0,\widetilde{T}]$ and $|N_{1}(t)|\leq
0.1r_{1}^{2}(t)$ for $t\in[\widetilde{T},T]$. This is possible, since
${\mathrm{D}}_{t}\log|N_{1}|<2{\mathrm{D}}_{t}\log r_{1}$. Then $|\mu(t)|\leq
C\sqrt{|N_{1}(t)|}$ for all $t\in[0,\widetilde{T}]$ and hence
$\int_{0}^{\widetilde{T}}|\mu(t)|{\mathrm{d}}t<C$.

##### Averaging Estimates.

Consider without loss of generality $t_{1},t_{2}\geq\widetilde{T}$. Using
$\Sigma_{+}\approx-1$ and $\delta_{1}\leq 0.1r_{1}$, we can estimate

$\mu\leq-0.4r_{1}^{2}\cos^{2}\psi+0.6r_{1}^{2}\sin^{2}\psi+0.01r_{1}^{2}+0.1r_{1}^{2}+C|N_{1}|\leq-0.25r_{1}^{2}+r_{1}^{2}\sin^{2}\psi.$

We can also estimate $\psi^{\prime}$:

$r_{1}|\sin\psi|\leq\psi^{\prime}\leq
2r_{1}\sqrt{\sin^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}.$

Let $\mu_{+}=r_{1}^{2}\sin^{2}\psi$ be the positive (bad) part of $\mu$; take
times $t_{1}<t_{L}<t_{R}<t_{2}$ with $|\psi(t_{R})-\psi(t_{L})|\leq 2\pi$.
Then

$\int_{t_{L}}^{t_{R}}\mu_{+}(t){\mathrm{d}}t\leq\int_{\psi(t_{L})}^{\psi(t_{R})}\frac{\mu_{+}(t)}{\psi^{\prime}(t)}{\mathrm{d}}t\leq
C_{r}^{-1}r_{1}(t_{R})\int_{0}^{2\pi}|\sin\psi|{\mathrm{d}}\psi\leq
C_{+}r_{1}(t_{R}),$

for some $C_{+}>0$. On the other hand, let $\mu_{-}=-0.25r_{1}^{2}$ be the
negative (good) part of $\mu$. Take times $t_{1}<t_{L}<t_{R}<t_{2}$ with
$\psi(t_{L})=k\pi-0.1$ and $\psi(t_{R})=k\pi+0.1$ for some $k\in{\mathbb{Z}}$;
then we can estimate

$\displaystyle\int_{t_{L}}^{t_{R}}\mu_{-}(t){\mathrm{d}}t$
$\displaystyle\leq\int_{-0.1}^{+0.1}\frac{-0.25\,r_{1}^{2}}{\psi^{\prime}}{\mathrm{d}}\psi\leq\int_{-0.1}^{+0.1}\frac{-0.125\,r_{1}}{\sqrt{\sin^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}}{\mathrm{d}}\psi$
$\displaystyle\leq-0.2\,C_{r}^{-1}r_{1}(t_{L})\int_{0}^{0.1}\frac{1}{\sqrt{x^{2}+C_{\delta,r}^{2}\epsilon_{v}^{2}}}{\mathrm{d}}x$

We can immediately see that for any $C_{+}>0$, we find $\epsilon_{v}>0$ such
that $\frac{\delta_{1}}{r_{1}}(t_{1})<\epsilon_{v}$ implies
$\int_{t_{L}}^{t_{R}}\mu_{-}(t){\mathrm{d}}t<-C_{+}r_{1}(t_{L})$. Hence, by
summing over $\psi$-rotations (and adjusting $\epsilon_{v}$), we can conclude
the assertion (5.3.2d). ∎

###### Proof of Proposition 5.3, conclusion (5.3.2e).

We need to show that solutions with small quotient $\frac{\delta_{1}}{r_{1}}$
cannot stay near ${\mathbf{T}}_{1}$ forever.

Assuming without loss of generality $|N_{1}|\ll r_{1}^{2}$ we have
${\mathrm{D}}_{\psi}\log r_{1}>Cr_{1}|\sin\psi|$. This shows that the only way
never leaving the vicinity of ${\mathbf{T}}_{1}$ is for the angle
$\psi\in{\mathbb{R}}$ to stay bounded, i.e.
$\lim_{t\to\infty}\psi(t)=\psi^{**}$ and $\lim_{t\to\infty}\delta_{1}(t)=0$
(since otherwise $r_{1}$ increases by a too large amount during each
rotation). This is impossible, since the possible limit-points lie on the
Kasner-circle $\mathcal{K}\setminus\\{{\mathbf{T}}_{1}\\}$ and are not
${\mathbf{T}}_{1}$; hence, either $N_{2}$ or $N_{3}$ is unstable and since
initially $N_{2}\neq 0\neq N_{3}$, the trajectory cannot converge to such a
point. ∎

###  5.4 Analysis in the Inaccessible-cones

Our whole approach aims at avoiding the much more tricky analysis of the
dynamics in the Inaccessible-cones, where possibly $\delta_{1}\geq r_{1}$:
Since trajectories starting outside of these cones never enter them, it is
unnecessary to know what happens in the Inaccessible-cones. However, for
various global questions, it is useful to collect at least some results inside
of these cones.

We already know that solutions in Bianchi IX cannot converge to the Taub-
points; the same holds in Bianchi VIII, even for solutions in Inaccessible:

######  Lemma 5.4.

For an initial condition ${\bm{x}}_{0}\in\mathcal{M}_{+-+}$, it is impossible
to have $\lim_{t\to\infty}{\bm{x}}(t)={\mathbf{T}}_{1}$.

###### Proof.

Suppose we have such a solution. Using equation (2.3.4), we can write near
${\mathbf{T}}_{1}$:

${\mathrm{D}}_{t}\log|N_{1}|\frac{\delta_{1}}{r_{1}}\leq\sqrt{3}\frac{|\Sigma_{-}|}{r_{1}}|N_{1}|\frac{\sqrt{N_{-}^{2}+\delta_{1}^{2}}}{r_{1}}-2.5.$

Hence, if ever $|N_{1}|\frac{\delta_{1}}{r_{1}}<1$, this inequality is
preserved and $|N_{1}|\frac{\delta_{1}}{r_{1}}$ decays exponentially. Then we
can estimate ${\mathrm{D}}_{t}\log
r_{1}\geq-C|N_{1}|-C|N_{1}|\frac{\delta_{1}}{r_{1}}$; all the terms on the
right hand side have bounded integral and $r_{1}\to 0$ is impossible.

On the other hand, if $r_{1}<|N_{1}|\delta_{1}$ for all sufficiently large
times, we can estimate ${\mathrm{D}}_{t}\log\delta_{1}\geq-
Cr_{1}^{2}-C|N_{1}|\geq-C|N_{1}|$, which has bounded integral and thus
contradicts $\delta_{1}\to 0$. ∎

Unfortunately, this is all we can presently say in the Inaccessible cone in
Bianchi VIII.

In the case of Bianchi IX, we can still average over $\psi$-rotations in order
to show that $\frac{\delta_{1}}{r_{1}}$ decays, even in the Inaccessible
region:

######  Lemma 5.5.

Let $h>0$. There exists constants $\epsilon,C>0$ such that the following
holds:

Let $\gamma:[0,T]\to\\{{\bm{x}}\in\mathcal{M}_{*++}:\,|N_{1}|\leq
r_{1}^{5},\,r_{1}<\epsilon,\,r_{1}\leq
h\delta_{1},\,d({\bm{x}},{\mathbf{T}}_{1})<0.1\\}$ be a partial trajectory
near ${\mathbf{T}}_{1}$. Then

$\log\frac{\delta_{1}}{r_{1}}(0)-\log\frac{\delta_{1}}{r_{1}}(T)\leq C(\log
r_{1}(T)-\log r_{1}(0)),$

i.e. the increase of $r_{1}$ and the decrease of $\frac{\delta_{1}}{r_{1}}$
have comparable rates.

###### Proof.

We can estimate

$\displaystyle{\mathrm{D}}_{t}\log r_{1}$
$\displaystyle=r_{1}^{2}\sin^{2}\psi\frac{-\Sigma_{+}}{1-\Sigma_{+}}+\mathcal{O}(|N_{1}|)$
$\displaystyle{\mathrm{D}}_{t}\log\frac{\delta_{1}}{r_{1}}$
$\displaystyle=\frac{-1}{1-\Sigma_{+}}r_{1}^{2}\cos^{2}\psi+\mathcal{O}(|N_{1}|)$
$\displaystyle\psi^{\prime}$
$\displaystyle=\sqrt{3}r_{1}\sqrt{\sin^{2}\psi+\frac{\delta_{1}^{2}}{r_{1}^{2}}}-\frac{r_{1}^{2}}{1-\Sigma_{+}}\cos\psi\sin\psi+\mathcal{O}(|N_{1}\sin\psi|).$

It suffices to show an estimate of the form

$\int_{0}^{2\psi}r_{1}^{2}\frac{\sin^{2}\psi}{\psi^{\prime}}{\mathrm{d}}\psi>C\int_{0}^{2\psi}\frac{r_{1}^{2}\cos^{2}\psi}{\psi^{\prime}}{\mathrm{d}}\psi.$

Using $\delta_{1}\geq hr_{1}$ and $r_{1}\leq\epsilon$, we can directly
estimate $\delta_{1}\leq\psi^{\prime}\leq 2\sqrt{r_{1}^{2}+\delta_{1}^{2}}$
(for $\epsilon>0$ small enough). This allows us to see that
$r_{1},\delta_{1},\frac{\delta_{1}}{r_{1}}$ can all change only by a bounded
factor during each rotation; hence it suffices to show for
$\psi(t_{2})-\psi(t_{1})\leq 2\pi$ that
$\min_{t\in[t_{1},t_{2}]}\psi^{\prime}(t)>C\max_{t\in[t_{1},t_{2}]}\psi^{\prime}(t).$

However,

$\min_{t\in[t_{1},t_{2}]}\psi^{\prime}(t)\geq\min_{t}\delta_{1}(t)\qquad\max_{t\in[t_{1},t_{2}]}\psi^{\prime}(t)\geq
2\max_{t}\sqrt{r_{1}^{2}(t)+\delta_{1}^{2}(t)}\leq\max_{t}\delta_{1}(t)\sqrt{1+h^{-2}},$

and we know that $\delta_{1}$ can only change by a bounded factor during each
rotation; hence the desired estimate follows. ∎

With a more subtle averaging argument than the previous ones, we can also show
the deferred Lemma 5.6. However, this proof is only given for the sake of
completeness and is nowhere used in this work, except for completing the
literature review in Section 3.3.

######  Lemma 5.6.

We consider without loss of generality the neighborhood of $\mathcal{T}_{1}$.
Let $\epsilon>0$ small enough. Then there exists a constant
$C_{\delta,\epsilon}\in(1,\infty)$, such that, for any piece of trajectory
$\gamma:[t_{1},t_{2}]\to\\{{\bm{x}}\in\mathcal{M}_{*++}:|{\bm{\Sigma}}({\bm{x}})-{\mathbf{T}}_{1}|\leq\epsilon,\,|N_{1}|\leq
10,\,\delta_{1}\leq 10\\}$, the following estimate holds:

$\delta_{1}(\gamma(t_{2}))\leq C_{\delta,\epsilon}\delta_{1}(\gamma(t_{1})).$

###### Proof.

We can assume without loss of generality that $\delta_{1}>hr_{1}$ for all
$t\in[t_{1},t_{2}]$ and for some $h=\hat{\epsilon}>0$; otherwise Proposition
5.3 applies. We can also assume without loss of generality that $|N_{1}|\leq
Cr_{1}^{2}$ for all $t\in[t_{1},t_{2}]$ because of a similar argument as in
the proof of Proposition 5.3.

We will use the letters $C\gg 1$ and $0<c\ll 1$ for unspecified constants.
Recall (2.3.2). We use the auxiliary variable
$\zeta=\frac{\delta_{1}}{r_{1}}\geq\hat{\epsilon}$. First, we note that

$\begin{array}[]{rccl}cr_{1}\zeta&\leq&\psi^{\prime}&\leq Cr_{1}\zeta\\\
cr_{1}^{2}\zeta^{-1}\cos^{2}\psi-C|N_{1}|\frac{1}{\psi^{\prime}}&\leq&{\mathrm{D}}_{\psi}r&\leq
Cr_{1}^{2}\zeta^{-1}\\\
cr_{1}\cos^{2}\psi-C|N_{1}|\frac{1}{\psi^{\prime}}&\leq&-{\mathrm{D}}_{\psi}\zeta&\leq
cr_{1}.\end{array}$

We see that $r_{1}$ and $\zeta$ can change only by a bounded factor (and also
a bounded amount) during a single rotation $[\psi_{1},\psi_{1}+2\pi]$;
therefore, we can average the equation for $r_{1}^{\prime}$ to see that