Abstract
In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the socalled Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the nonuniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a nonuniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.
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Notes
A Lie group is called unimodular if its left invariant Haar measure is also right invariant.
More precisely, all Bianchi spacetimes, except the simplest ones, namely Minkowski spacetimes and TaubNUT spacetimes.
We say that a maximal vacuum class A Bianchi spacetime \((M,g)=(I\times G, \textrm{d}s^2+h_s)\) admits an initial singularity if \(I=]s_{},s_{+}[\) with \(s_{} >  \infty \). If this is the case, the curvature blows up when the time tends to \(s_{}\) (see [Rin00]).
It is denoted by \(\tau \) in [HU09] (they choose to respect the “physical” timeorientation).
Here, Bianchi spacetimes are considered up to isomorphism, metric rescaling, time orientation reversal and time translation. Maximal solutions of the Wainwright–Hsu equations are considered up to permutation of the indices 1, 2, 3, simultaneous sign reversal of the \(N_i\)’s and time translation. The Minkowski spacetime does not correspond to any solution of the Wainwright–Hsu equations. Each Bianchi spacetime of type \({{\,\textrm{IX}\,}}\) (see paragraph 1.1.4) splits into two halves (the expanding part and the contracting part), each of which correspond to a solution of the Wainwright–Hsu equation.
Precisely, the \(\omega \)limit set of an orbit \({\mathscr {O}}(t)\) is defined as the set \(\omega ({\mathscr {O}}) \overset{\text {def}}{=}\cap _{s \ge 0}\overline{\{ {\mathscr {O}}(t) \,  \, t \ge s \}}\). If \({\mathscr {O}}\) converges to a point x in the future, then \(\omega ({\mathscr {O}})= \{x\}\) and we say that x is the \(\omega \)limit point of \({\mathscr {O}}\).
This formulation is classic and is based on the work of Beliinski, Khalatnikov and Lifschitz on one hand and Misner on the other hand.
A heteroclinic connexion is an orbit “joining two different points”. More precisely it is an orbit \(t \mapsto {\mathscr {O}}(t)\) such that there exists two distinct points p and q verifying \(\lim _{t \rightarrow +\infty } {\mathscr {O}}(t) = q\) and \(\lim _{t \rightarrow \infty } {\mathscr {O}}(t) = p\).
Strictly speaking, the Kasner map does not satisfy the hypothesis [KH97, Theorem 2.4.6], since the modulus of its derivative is not bounded from below by a constant \(\mu >1\). Nevertheless, one can easily check that the proof of [KH97, Theorem 2.4.6] still works for nonuniformly expanding maps such as the Kasner map.
The word smooth will always stand for \(C^{\infty }\) in this work.
Any connected component of \({\mathcal {O}}_{\omega }^{u} \cap {\mathcal {U}}_{\xi }\) is oriented by the flow of the Wainwright–Hsu vector field \({\mathcal {X}}\).
We would like to thank Sébastien Gouëzel for explaining to us how to use the PerronFrobenius operator here.
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Acknowledgements
The authors express their sincere gratitude to the referees, who spent many hours carefully reading previous versions of the paper, made important suggestions to clarify some parts of it, and pointed out some errors and inaccuracies.
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This article is based on the Ph.D. thesis of the second author, under the supervision of the first one. We wish to thank the Laboratoire d’Analyse, Géométrie et Applications, where this thesis was conducted. We also wish to thank Jérôme Buzzi and Hans Ringström who wrote a report on this thesis and suggested many improvements.
Appendices
Continued Fractions
In this appendix, we gather the results about continued fractions that are used in the memoir. The main result is Lemma 1.8. We also prove a result on the expansivity of the Gauss transformation.
We first need to introduce some notations. Set \(\Omega =[0,1] {\setminus } {\mathbb {Q}}\). For every \(x \in \Omega \), there exists a unique sequence \((k_n(x))_{n \ge 1}\) of integers larger than 1 such that \(x=\lim _{n \rightarrow +\infty } [k_1(x),\dots ,k_n(x)]\) where
We use the notation
Lemma 1.8 is a straightforward consequence of the following lemma.
Lemma A.1
For Lebesgue almost every \(x \in \Omega \), there exists \(n_0 \in {\mathbb {N}}\) such that for every \(n \ge n_0\),
and
The inequality (A.2) is a consequence of a standard fact: for Lebesgue almost every point \(x \in \Omega \), the sequence \((k_i(x))_{i \ge 0}\) of the partial quotients of x does not grow “too fast” (see corollary A.3). The inequality (A.1) is a consequence of a less standard result: for Lebesgue almost every point \(x \in \Omega \) and for every \(n \in {\mathbb {N}}\) large enough, there is at least one partial quotient among \(k_1(x)\), ..., \(k_n(x)\) which is “large” (see Proposition A.4). More precisely, the standard result can be rigorously stated as follows.
Proposition A.2
Let \(\varphi : {\mathbb {N}}^* \rightarrow {\mathbb {R}}^*_+\). Either the set
is finite for Lebesgue almost all \(x \in \Omega \), or it is infinite for Lebesgue almost all \(x \in \Omega \). More precisely, this dichotomy depends on \(\varphi \) as follows:

1.
If \(\sum \frac{1}{\varphi (n)}\) is divergent, then for Lebesgue almost all \(x \in \Omega \), there exists infinitely many \(n \in {\mathbb {N}}^*\) such that \(k_n(x) \ge \varphi (n)\).

2.
If \(\sum \frac{1}{\varphi (n)}\) is convergent, then for Lebesgue almost all \(x \in \Omega \), there exists \(n_0(x) \in {\mathbb {N}}^*\) such that for every \(n \ge n_0(x)\), \(k_n(x) < \varphi (n)\).
Proof
See [Khi64]. \(\square \)
Corollary A.3
Let \(\epsilon >0\). For Lebesgue almost every point \(x \in \Omega \), there exists \(n_0 \in {\mathbb {N}}\) such that for every \(n \ge n_0\), \(k_n(x) \le n^{1+\epsilon }\).
Proof
For any \(\epsilon >0\), the serie \(\sum n^{1\epsilon }\) is convergent. \(\square \)
We now give a precise formulation of the second result needed to prove Lemma A.1.
Proposition A.4
For Lebesgue almost all \(x \in \Omega \), for every \(\epsilon >0\), there exists \(n_0(x,\epsilon ) \ge 1\) such that for every \(n \ge n_0(x,\epsilon )\), there exists an integer \(1 \le j \le n\) such that \(k_j(x) \ge n^{1\epsilon }\).
Let us introduce some tools that will be needed to prove Proposition A.4. We denote by \(\tau :\Omega \rightarrow \Omega \) the Gauss transformation defined by \(\tau (x) = \left\{ \frac{1}{x}\right\} \) where \(\left\{ x\right\} =x\lfloor x \rfloor \) denotes the fractional part of x. The very definition of \(\tau \) implies that, for every continued fraction \([k_1,k_2,\dots ]\),
In other words, \(\tau \) is conjugated to the left shift on the space of sequences \((k_n)_{n \ge 1}\) of integers larger than 1.
Let us denote by \(\gamma _G\) the Gauss measure, defined by
where \(\lambda \) denotes the Lebesgue measure. One can remark that the Gauss measure \(\gamma _G\) is equivalent to the Lebesgue measure \(\lambda \) on [0, 1]. The fundamental fact is that \(\gamma _G\) is \(\tau \)invariant, i.e.
For any map \(f:[0,1]\rightarrow {\mathbb {C}}\), let
where the supremum is taken on all the finite sequences \(0\le t_1< \dots < t_n \le 1\), \(n \ge 2\). If \({{\,\textrm{var}\,}}f < +\infty \), then we say that f is of bounded variation. For any map \(f \in L_\lambda ^\infty ([0,1])\), we call essential variation off and we denote by v(f) the number \(\inf {{\,\textrm{var}\,}}{\tilde{f}}\) where the infimum is taken on all the maps \({\tilde{f}}\) equal to f mod 0. If \(v(f)< +\infty \), then we say that f is of bounded essential variation. Let us denote by \({{\,\textrm{BEV}\,}}([0,1])\) the set of all maps \(f \in L_\lambda ^\infty ([0,1])\) such that \(v(f) < +\infty \). Let us equip \({{\,\textrm{BEV}\,}}([0,1])\) with the norm
We define the PerronFrobenius operator \(U\) as the “dual” of the composition operator induced by \(\tau \). More precisely, \(U\) is defined as the unique bounded linear operator \(L_\lambda ^1([0,1]) \rightarrow L_\lambda ^1([0,1])\) satisfying, for every \(f \in L_\lambda ^1([0,1])\) and for every \(g \in L_\lambda ^\infty ([0,1])\),
Proposition A.5
(Spectral gap for the PerronFrobenius operator). The PerronFrobenius operator has a spectral gap: there exists \(0<\alpha <1\) and \(C >0\) such that, for every \(f \in {{\,\textrm{BEV}\,}}([0,1])\),
Proof
See [IK13]. \(\square \)
Proof of Proposition A.4
^{Footnote 13} Let us define, for \(n \ge 2\) and \(\epsilon >0\) small:
According to the Borel–Cantelli lemma, it is enough to prove that
One can remark that
so
Let \(c=\left\lfloor n^{\frac{\epsilon }{2}} \right\rfloor \) and \(K=\left\lfloor \frac{n1}{n^{\frac{\epsilon }{2}}} \right\rfloor \). We can estimate the above integral by keeping only the terms whose indices are multiples of c:
However, the family \(\left( {\textbf{1}}_{X_{n,\epsilon }}\right) _n\) is uniformly bounded by 2 in \({{\,\textrm{BEV}\,}}([0,1])\) and \(\prod _{j=0}^{K1} {\textbf{1}}_{X_{n,\epsilon }}\circ \tau ^{jc}\) is bounded by 1 in \(L_\lambda ^\infty ([0,1])\) so according to the Proposition A.5,
By induction, we get
However, \(X_{n,\epsilon } = \Omega \cap ]\frac{1}{\lfloor n^{1\epsilon } \rfloor +1},1 [\) and using (A.3), we get that
Moreover,
Hence,
and \(\gamma _G\left( X_{n,\epsilon } \right) ^{K+1}\) is the general term of a convergent series. Analogously, \(K\alpha ^c\) is the general term of a convergent series. Hence, (A.4) holds true. This concludes the proof of Proposition A.4. \(\square \)
Proof of Lemma A.1
Inequalities (A.1) and (A.2) are straightforward consequences of corollary A.3 and Proposition A.4 respectively, with \(\epsilon =10^{2}\). \(\square \)
The following result provides some explicit conditions ensuring that the continued fraction expansion of two nearby real numbers start by the same integer. It is used to prove Lemma 8.8. In particular, it is useful to find a sufficiently small size for the section \(S_{\omega ,\textbf{h}_{\omega }}^{s}\) so that all the points (in fact, their coordinate \(x_c\)) in \(S_{\omega ,\textbf{h}_{\omega }}^{s}\) have the same first partial quotient.
Proposition A.6
For \(x,x' \in \Omega \), if
then \(k_1(x')=k_1(x)\).
Proof
Fix \(x=[k_1,k_2,\dots ] \in \Omega \). Let \(x'=[k_1',k_2',\dots ] \in \Omega \) such that
One can remark that
By a straightforward computation, one gets
and
It follows that
Hence, \(k_1'=k_1\). \(\square \)
The following results provide some explicit conditions ensuring that the continued fraction expansion of two nearby real numbers start by the same first two integers. Moreover, it shows that the double Gauss transformation \(\tau ^2\) is expansive. It is particularly useful to prove Lemma 9.11.
Proposition A.7
(Expansivity of \(\tau ^2\)). For \(x,x' \in \Omega \), if
then \(k_1(x')=k_1(x)\), \(k_2(x')=k_2(x)\) and
Proof
Fix \(x=[k_1,k_2,\dots ] \in \Omega \). Let \(x'=[k_1',k_2',\dots ] \in \Omega \) such that
One can remark that
By a straightforward computation, one gets
and
It follows that
Hence, \(k_1'=k_1\) and \(k_2'=k_2\). Writing
leads to
Since \(k_1k_2+1 \ge 2\), we get
\(\square \)
Statement of the Main Theorem in the Entire Phase Space
In this appendix, we explain how to extend Theorem 1.9 to type \({{\,\textrm{VIII}\,}}\) orbits. To this end, we show how some objects defined in the introduction (especially type \({{\,\textrm{II}\,}}\) orbits, the Kasner map and heteroclinic chains) can be generalized to the entire phase space. A technical complication arises since most abstract heteroclinic chains cannot be shadowed by any type \({{\,\textrm{VIII}\,}}\) or \({{\,\textrm{IX}\,}}\) orbit for elementary reasons. This will lead us to introduce a notion of coherent heteroclinic chain.
Type \({{\,\textrm{II}\,}}\) orbits. Recall that in \({\mathscr {B}}^+\), for every point p of the Kasner circle that is not a Taub point, there is exactly one type \({{\,\textrm{II}\,}}\) orbit starting at p. When looking at the full phase space \({\mathscr {B}}\), we have the following result. For every point p of the Kasner circle that is not a Taub point, there are exactly two type \({{\,\textrm{II}\,}}\) orbits starting at p. These two orbits are exchanged by the symmetry
fixing the points of the plane \(\left( N_1=N_2=N_3=0\right) \) containing the Kasner circle. As an immediate consequence, these two type \({{\,\textrm{II}\,}}\) orbits converge to the same point of \({\mathscr {K}}\) in the future.
Kasner map Let p be a point of the Kasner circle which is not a Taub point. When we restrict ourselves to \({\mathscr {B}}^+\), there is exactly one type \({{\,\textrm{II}\,}}\) orbit starting at p and this orbit converges to a point denoted by \({\mathscr {F}}(p)\) (the image of p by the Kasner map). This is indeed how we defined the Kasner map (see section 2.5). As stated above, in \({\mathscr {B}}\), there are two (symmetrical) type \({{\,\textrm{II}\,}}\) orbits starting at p. Since they are symmetrical, they both converge to the same point of the Kasner circle, that is, the point \({\mathscr {F}}(p)\). We will denote these two type \({{\,\textrm{II}\,}}\) orbits by \({\mathscr {O}}_{p \rightarrow {\mathscr {F}}(p)}^{+}\) and \({\mathscr {O}}_{p \rightarrow {\mathscr {F}}(p)}^{}\), \({\mathscr {O}}_{p \rightarrow {\mathscr {F}}(p)}^{+}\) being the one entirely contained in \({\mathscr {B}}^+\).
Coherent heteroclinic chains
Definition B.1
(Heteroclinic chains). Let p be a point of the Kasner circle (such that, for every \(k \ge 0\), \({\mathscr {F}}^k(p)\) is not a Taub point). A heteroclinic chain (starting at p) is a concatenation of one type \({{\,\textrm{II}\,}}\) orbit starting at p and arriving at \({\mathscr {F}}(p)\), then one type \({{\,\textrm{II}\,}}\) orbit starting at \({\mathscr {F}}(p)\) and arriving at \({\mathscr {F}}^2(p)\), etc. Formally, this is a sequence of the form
where \(\epsilon _n \in \{ \pm \}\) corresponds to a choice of one of the two symmetrical type \({{\,\textrm{II}\,}}\) orbits starting at \({\mathscr {F}}^n(p)\).
As we will see, some heteroclinic chains cannot be shadowed by type \({{\,\textrm{VIII}\,}}\) or type \({{\,\textrm{IX}\,}}\) orbits. First, let us recall the definition of shadowing, generalized to the full phase space in a straightforward manner.
Definition B.2
(Shadowing). Let \(t \mapsto {\mathscr {O}}(t)\) be a type \({{\,\textrm{VIII}\,}}\) or \({{\,\textrm{IX}\,}}\) orbit in \({\mathscr {B}}\), p be a point of the Kasner circle (such that, for every \(k \ge 0\), \({\mathscr {F}}^k(p)\) is not a Taub point) and \({\mathscr {H}}\) be a heteroclinic chain (B.1) starting at p. We say that \({\mathscr {O}}\) shadows \({\mathscr {H}}\) (or \({\mathscr {H}}\) attracts \({\mathscr {O}}\)) if there exists a strictly increasing sequence \((t_n)_{n \in {\mathbb {N}}} \subset {\mathbb {R}}_+\) such that

1.
\(d({\mathscr {O}}(t_n),{\mathscr {F}}^n(p)) \xrightarrow [n\rightarrow +\infty ]{} 0\).

2.
The Hausdorff distance between the orbit interval \(\{ {\mathscr {O}}(t) \,  \, t_n< t < t_{n+1}\}\) and the type \({{\,\textrm{II}\,}}\) orbit \({\mathscr {O}}_{{\mathscr {F}}^{n}(p) \rightarrow {\mathscr {F}}^{n+1}(p)}^{\epsilon _n}\) tends to 0 when \(n \rightarrow + \infty \).
Recall that any type \({{\,\textrm{II}\,}}\) orbit is contained in a subset of the phase space of the form
where \(\{i,j,k\} = \{1,2,3\}\). Consider for example a heteroclinic chain made of an infinite number of type \({{\,\textrm{II}\,}}\) orbits traveling in \(\{ N_1 >0, N_2 =0, N_3 =0 \}\) and an infinite number of type \({{\,\textrm{II}\,}}\) orbits traveling in \(\{ N_1 <0, N_2 =0, N_3 =0 \}\). Let \(t \mapsto {\mathscr {O}}(t)=\left( N_1(t),N_2(t),N_3(t),\Sigma _1(t),\Sigma _2(t),\Sigma _3(t)\right) \) be a type \({{\,\textrm{VIII}\,}}\) or \({{\,\textrm{IX}\,}}\) orbit. Recall that the signs of the variables \(N_i\) are constant. Hence, it is obvious that \({\mathscr {O}}\) cannot shadow this heteroclinic chain, as it would violate the fact that the sign of \(N_1\) is constant along \({\mathscr {O}}\). This means that any heteroclinic chain “alternating” between two signs as in the example above has zero chance to attract some type \({{\,\textrm{VIII}\,}}\) or \({{\,\textrm{IX}\,}}\) orbits.
This leads us to the definition of coherent heteroclinic chains. Recall that the Mixmaster attractor is the union of three ellipsoids and each of these ellipsoids is the union of two symmetrical hemiellipsoids (they correspond to opposite signs for one of the three variables \(N_i\)). In other words,
where
and analogously for the other hemiellipsoids.
Definition B.3
(Coherent heteroclinic chain). A heteroclinic chain of type \({{\,\textrm{II}\,}}\) orbits is coherent if it is included in the union of three hemiellipsoids (in three different directions) bounded by the Kasner \({\mathscr {K}}\), that is, if it is included in a set of the form
For every point p of the Kasner circle (such that, for every \(k \ge 0\), \({\mathscr {F}}^k(p)\) is not a Taub point), there are exactly eight coherent heteroclinic chains starting at p corresponding to the eight different choices of three hemiellipsoids (or, analogously, corresponding to the eight different choices of three signs for the variables \(N_i\)). One should remark that a type \({{\,\textrm{VIII}\,}}\) orbit cannot shadow the same coherent heteroclinic chain as a type \({{\,\textrm{IX}\,}}\) orbit. Among the eight coherent heteroclinic chains starting at p, six can be shadowed by type \({{\,\textrm{VIII}\,}}\) orbits and two by type \({{\,\textrm{IX}\,}}\) orbits.
Having this definition in mind, it is clear that Theorem 1.9 must be generalized by replacing the unique heteroclinic chain in \({\mathscr {B}}^+\) starting at p by one of the eight coherent heteroclinic chains in \({\mathscr {B}}\) starting at p. Recall that \({\mathscr {K}}_{(MG)}\) denotes the set of all the points \(p \in {\mathscr {K}}\) such that \(\omega (p)\) verifies the moderate growth condition (MG).
Main theorem B.4
Let p be a point of the Kasner circle and let \({\mathscr {H}}\) be a coherent heteroclinic chain starting at p. If \(\omega (p)\) verifies the moderate growth condition (MG), then the union of all the type \({{\,\textrm{VIII}\,}}\) or \({{\,\textrm{IX}\,}}\) orbits shadowing the heteroclinic chain \({\mathscr {H}}\) contains a 3dimensional ball \(D(p,{\mathscr {H}})\) Lipschitz embedded in the phase space \({\mathscr {B}}^+\). Moreover, for any \({\mathscr {E}}\subset {\mathscr {K}}_{(MG)}\) of positive 1dimensional Lebesgue measure, the union of all the balls \(D(p,{\mathscr {H}})\) for \(p \in {\mathscr {E}}\) and \({\mathscr {H}}\) a coherent heteroclinic chain starting at p has positive 4dimensional Lebesgue measure.
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Béguin, F., Dutilleul, T. Chaotic Dynamics of Spatially Homogeneous Spacetimes. Commun. Math. Phys. 399, 737–927 (2023). https://doi.org/10.1007/s00220022045838
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DOI: https://doi.org/10.1007/s00220022045838