aa r X i v : . [ g r- q c ] J a n Mixmaster: Fact and Belief
J. Mark Heinzle ∗ Gravitational Physics, Faculty of Physics,University of Vienna, A-1090 Vienna, AustriaandMittag-Leffler Institute of the Royal Swedish Academy of SciencesS-18260 Djursholm, Sweden
Claes Uggla † Department of Physics,University of Karlstad, S-651 88 Karlstad, Sweden
Abstract
We consider the dynamics towards the initial singularity of Bianchi type IX vacuumand orthogonal perfect fluid models with a linear equation of state. Surprisingly few factsare known about the ‘Mixmaster’ dynamics of these models, while at the same time most ofthe commonly held beliefs are rather vague. In this paper, we use Mixmaster facts as a baseto build an infrastructure that makes it possible to sharpen the main Mixmaster beliefs.We formulate explicit conjectures concerning (i) the past asymptotic states of type IXsolutions and (ii) the relevance of the Mixmaster/Kasner map for generic past asymptoticdynamics. The evidence for the conjectures is based on a study of the stochastic propertiesof this map in conjunction with dynamical systems techniques. We use a dynamical systemsformulation, since this approach has so far been the only successful path to obtain theorems,but we also make comparisons with the ‘metric’ and Hamiltonian ‘billiard’ approaches. ∗ Electronic address: [email protected] † Electronic address: [email protected] INTRODUCTION Today, Bianchi type IX enjoys an almost mythical status in general relativity and cosmology,which is due to two commonly held beliefs: (i) Type IX dynamics is believed to be essentiallyunderstood; (ii) Bianchi type IX is believed to be a role model that captures the generic featuresof generic spacelike singularities. However, we will illustrate in this paper that there are reasonsto question these beliefs.The idea that type IX is essentially understood is a misconception. In actuality, surprisinglylittle is known, i.e., proved, about type IX asymptotic dynamics; at the same time there existwidely held, but rather vague, beliefs about Mixmaster dynamics, oscillations, and chaos, whichare frequently mistaken to be facts. There is thus a need for clarification: What are the knownfacts and what is merely believed about type IX asymptotics? We will address this issue in twoways: On the one hand, we will discuss the main rigorous results on Mixmaster dynamics, the‘Bianchi type IX attractor theorem’, and its consequences; in particular, we will point out thelimitations of these results. On the other hand, we will provide the infrastructure that makes itpossible to sharpen commonly held beliefs; based on this framework we will formulate explicitrefutable conjectures.Historically, Bianchi type IX vacuum and orthogonal perfect fluid models entered the scenein the late sixties through the work of Belinskii, Khalatnikov and Lifshitz [1, 2] and Misnerand Chitr´e [3, 4, 5, 6]. BKL attempted to understand the detailed nature of singularities andwere led to the type IX models via a rather convoluted route, while Misner was interested inmechanisms that could explain why the Universe today is almost isotropic. BKL and Misnerindependently, by means of quite different methods, reached the conclusion that the temporalbehavior of the type IX models towards the initial singularity can be described by sequences ofanisotropic Kasner states, i.e., Bianchi type I vacuum solutions. These sequences are determinedby a discrete map that leads to an oscillatory anisotropic behavior, which motivated Misner torefer to the type IX models as Mixmaster models [3, 4]. This discrete map, the Kasner map, waslater shown to be associated with stochasticity and chaos [7, 8, 9], a property that has generatedconsiderable interest—and confusion, see, e.g., [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] andreferences therein. A sobering thought: All claims about chaos in Einstein’s equations reston the (plausible) belief that the Kasner map actually describes the asymptotic dynamics ofEinstein’s equations; as will be discussed below, this is far from evident (despite being plausible)and has not been proved so far.More than a decade after BKL’s and Misner’s investigations a new development took place:Einstein’s field equations in the spatially homogeneous (SH) case were reformulated in a mannerthat allowed one to apply powerful dynamical systems techniques [21, 22, 23]; gradually a pictureof a hierarchy of invariant subsets emerged where monotone functions restricted the asymptoticdynamics to boundaries of boundaries, see [10] and references therein. Based on work reviewedand developed in [10] and by Rendall [24], Ringstr¨om eventually produced the first major proofsabout asymptotic type IX dynamics [25, 26]. This achievement is remarkable, but it does notfollow that all questions are settled. On the contrary, so far nothing is rigorously known, e.g.,about dynamical chaotic properties (although there are good grounds for beliefs), nor has therole of type IX models in the context of generic singularities been established [27, 28, 29, 30].The outline of the paper is as follows. In Section 2 we briefly describe the Hubble-normalizeddynamical systems approach and establish the connection with the metric approach. For sim-plicity we restrict ourselves to the vacuum case and the so-called orthogonal perfect fluid case,i.e., the fluid flow is orthogonal w.r.t. the SH symmetry surfaces; furthermore, we assume alinear equation of state. In Section 3 we discuss the levels of the Bianchi type IX so-called We will refer to the authors and their work as BKL.
BASIC EQUATIONS c = 1 and 8 πG = 1, where c is the speed of light and G the gravitationalconstant. We consider vacuum or orthogonal perfect fluid SH Bianchi type IX models (i.e., the fluid 4-velocity is assumed to be orthogonal to the SH symmetry surfaces) with a linear equation ofstate; we require the energy conditions (weak/strong/dominant) to hold, i.e., ρ > − < w < , (1)where w = p/ρ , and where ρ and p are the energy density and pressure of the fluid, respectively.By (1) we exclude the special cases w = − and w = 1, where the energy conditions are onlymarginally satisfied. As is well known, see, e.g., [10, 26] and references therein, for these models there exists asymmetry-adapted (co-)frame { ˆ ω , ˆ ω , ˆ ω } , d ˆ ω = − ˆ n ˆ ω ∧ ˆ ω , d ˆ ω = − ˆ n ˆ ω ∧ ˆ ω , d ˆ ω = − ˆ n ˆ ω ∧ ˆ ω , (2a)with ˆ n = 1, ˆ n = 1, ˆ n = 1, such that the type IX metric takes the form g = − dt ⊗ dt + g ( t ) ˆ ω ⊗ ˆ ω + g ( t ) ˆ ω ⊗ ˆ ω + g ( t ) ˆ ω ⊗ ˆ ω . (2b) Note that the well-posedness of the Einstein equations (for solutions without symmetry) has been questionedin the case − / < w <
0, see [32]. The case w = 1 is known as the stiff fluid case, for which the speed of soundis equal to the speed of light. The asymptotic dynamics of stiff fluid solutions is simpler than the oscillatorybehavior characterizing the models with range − < w <
1, and well understood [26, 33]. (In the terminologyintroduced below, the stiff fluid models are asymptotically self-similar.) We will therefore refrain from discussingthe stiff fluid case in this paper.
BASIC EQUATIONS n α ˆ n β ˆ n γ I 0 0 0II 0 0 +VI − +VII − + +IX + + +Table 1: The class A Bianchi types are characterized by different signs of the structure constants(ˆ n α , ˆ n β , ˆ n γ ), where ( αβγ ) is any permutation of (123). In addition to the above representationsthere exist equivalent representations associated with an overall change of sign of the structureconstants; e.g., another type IX representation is ( − − − ).Hence, the type IX models naturally belong to the so-called class A Bianchi models, see Table 1.Let n ( t ) := ˆ n g √ det g , n ( t ) := ˆ n g √ det g , n ( t ) := ˆ n g √ det g , (3)where det g = g g g . Furthermore, define θ = − tr k and σ αβ = − k αβ + tr k δ αβ = diag( σ , σ , σ ) (cid:16) ⇒ X α σ α = 0 (cid:17) , (4)where k αβ denotes the second fundamental form associated with (2) of the SH hypersurfaces t = const. The quantities θ and σ αβ can be interpreted as the expansion and the shear,respectively, of the normal congruence of the SH hypersurfaces. In a cosmological context itis customary to replace θ by the Hubble variable H = θ/ − tr k/
3; this variable is relatedto changes of the spatial volume density according to d √ det g/dt = 3 H √ det g . Evidently, inBianchi type IX (and type VIII) there is a one-to-one correspondence between the ‘orthonormalframe variables’ ( H, σ α , n α ) (with P α σ α = 0) and ( g αβ , k αβ ); in particular, the metric g αβ isobtained from ( n , n , n ) via (3). (For the lower Bianchi types I–VII , some of the variables( n , n , n ) are zero, cf. (3); in this case, the other frame variables, i.e., ( H, σ α ), are needed aswell to reconstruct the metric; see [34] for a group theoretical approach.)In the Hubble-normalized dynamical systems approach we define dimensionless orthonormalframe variables according to(Σ α , N α ) = ( σ α , n α ) /H , Ω = ρ/ (3 H ) . (5)In addition we introduce a new dimensionless time variable τ according to dτ /dt = H . Like thecosmological time t , the time τ is directed towards the future; however, to make contact withthe well established convention that uses a past-directed ‘time’ for the discrete Mixmaster map,see Section 5, it will occasionally become necessary to use an inverse time τ − = − τ instead of τ itself.For all class A models except type IX the Gauss constraint guarantees that H remains positiveif it is positive initially. In Bianchi type IX, however, it is known from a theorem by Lin andWald [35] that all type IX vacuum and orthogonal perfect fluid models with w ≥ H > H = 0), and then recollapse ( H < Thevariable transformation (5) breaks down at the point of maximum expansion in the type IXcase; however, the variables (Σ α , N α ) correctly describe the dynamics in the expanding phase,which we will focus on henceforth. In the locally rotationally symmetric case it has been proved that the range of w can be extended to w > − ,see [36]. There are good reasons to believe that the assumption of local rotational symmetry is superfluous, butthis has not been established yet. BASIC EQUATIONS H and the Hubble-normalizedvariables (Σ α , N α ) it follows for dimensional reasons that the equation for the single variablewith dimension, H , H ′ = − (1 + q ) H , (6)decouples from the remaining dimensionless equations [10]; here and henceforth a prime denotesthe derivative d/dτ . The equations for (Σ α , N α ) form the following coupled system [10]:Σ ′ α = − (2 − q )Σ α − S α , (7a) N ′ α = ( q + 2Σ α ) N α (no sum over α ) , (7b)where q = 2Σ + (1 + 3 w )Ω , Σ = (Σ + Σ + Σ ) , (8a)and S α = (cid:2) N α (2 N α − N β − N γ ) − ( N β − N γ ) (cid:3) , ( αβγ ) ∈ { (123) , (231) , (312) } . (8b)Note that the ‘deceleration parameter’ q is non-negative because of the assumption w > − / + Σ + Σ = 0, there exists the Gauss constraintΣ + h N + N + N − N N + N N + N N ) | {z } ∆ II i + Ω = 1 , (9)which is used to globally solve for Ω when Ω = 0. Accordingly, the reduced state space is givenas the space of all (Σ , Σ , Σ ) and ( N , N , N ) such that Σ + Σ + Σ = 0 andΣ + h N + N + N − N N + N N + N N ) i ≤ , (9 ′ )which follows from (9) under the assumption that Ω ≥ ρ ≥ while the state space of the vacuum models (i.e., Ω = 0) is4-dimensional. The same is true for Bianchi type VIII, while the state spaces of the remainingclass A Bianchi models have less degrees of freedom; see Table 2. Once the dynamics in thedimensionless state space is understood, H is obtained from a quadrature by integrating (6),which allows one to reconstruct the metric.For all class A models except type IX the constraint (9 ′ ) implies that Σ ≤
1; in Bianchitype IX, however, Σ > ( N N N ) / . (10)Employing (8a) and (9) and using that h N + N + N − II i + ∆ ≥ , (11)where equality holds iff N = N = N , we findΣ ≤ , Ω ≤ , − q ≥ (1 − w )Ω − . (12)The function ∆ is strictly monotonically increasing along orbits of Bianchi type IX. To see thiswe use (7) and compute∆ ′ = 2 q ∆ , ∆ ′′ (cid:12)(cid:12)(cid:12) q =0 = 0 , ∆ ′′′ (cid:12)(cid:12)(cid:12) q =0 = (cid:2) S + S + S (cid:3) ∆ , (13) It is common to globally solve Σ + Σ + Σ = 0 by introducing new variables according to Σ = − + ,Σ = Σ + − √ − , Σ = Σ + + √ − , which yields Σ = Σ + Σ − . However, since this breaks the permutationsymmetry of the three spatial axes (exhibited by type IX models), we choose to retain the variables Σ , Σ , Σ . THE BIANCHI TYPE IX LIE CONTRACTION HIERARCHY N α , N β , N γ ) State space D I B I N α = 0, N β = 0, N γ = 0 Σ ≤ B II N α = 0, N β = 0, N γ > + N γ ≤ B VI N α = 0, N β < N γ > + [ N β − N γ ] ≤ B VII N α = 0, N β > N γ > + [ N β − N γ ] ≤ B VIII N α < N β > N γ > ′ ) ( ⇒ Σ <
1) 5IX B IX N α > N β > N γ > ′ ) ( ⇒ Σ ≤ αβγ )is any permutation of (123). In addition to the above representations there exist equivalentrepresentations associated with an overall change of sign of the variables ( N , N , N ). Thequantity D denotes the dimension of the state space (in the fluid case); the dimensionality ofthe state space in the vacuum cases is given by D − S + S + S > and Ω are bounded towards the past .The right hand side of the reduced system (7) consists of polynomials of the state space variablesand is thus a regular dynamical system. Solutions of (7) of Bianchi types I–VIII are global in τ , since (9) implies the bounds Σ ≤ ≤ N α in (7b).Solutions of (7) of Bianchi type IX are global towards the past, since q + 2Σ α is bounded frombelow; this follows from (8a), which yields q ≥ , so that q + 2Σ α ≥ − (Σ α + 2) + (Σ β − Σ γ ) . The decoupled equation for H , cf. (6), yields that H → ∞ as τ → −∞ , because q is non-negative. Since q is bounded as τ → −∞ , the asymptotics of H can be bounded byexponential functions from above and below. It follows that the equation dt/dτ = H − can beintegrated to yield t as a function of τ such that t → τ → −∞ .In addition to (7) it is useful to also consider an auxiliary equation for the matter quantity Ω,Ω ′ = [2 q − (1 + 3 w )]Ω . (14)Making use of (5) and (6) we conclude that for all orthogonal perfect fluid models with a linearequation of state we have ρ ∝ exp ( − w ] τ ), and hence ρ → ∞ as τ → −∞ , which yieldsa past singularity. The divergence of ρ can also be directly read off from the matter equation ∇ a T ab = 0. The Bianchi type IX state space B IX is characterized by the conditions N > N > N > B IX = B N N N . The notation is such that the subscript denotes the non-zero variablesamong { N , N , N } . Setting one or more of these variables to zero (which corresponds to Liecontractions [37]) yields invariant boundary subsets which describe more special Bianchi types.Since the type IX models exhibit discrete symmetries associated with axes permutations, thecontractions generate all possible representations of the more special (Lie contracted) Bianchitypes (which are associated with such permutations): The Bianchi type VII subspace B VII is given by the disjoint union of three equivalent sets, B VII = B N N ∪ B N N ∪ B N N , where,e.g., B N N denotes the type VII subset with N > N > N = 0; the Bianchi type IIsubspace B II by the union B II = B N ∪ B N ∪ B N ; the Bianchi type I subspace B I by B I = B ∅ .Note that the Bianchi type VI subspace does not appear as a boundary subset of B IX . ABianchi subset contraction diagram for type IX is given in Figure 1. THE BIANCHI TYPE IX LIE CONTRACTION HIERARCHY D2345
PSfrag replacements B N N N B N N B N N B N N B N B N B N B ∅ B IX B VII B II B I Figure 1: Subset contraction diagram for Bianchi type IX. D denotes the dimension of thedimensionless state space for the various models with an orthogonal perfect fluid with linearequation of state; the associated vacuum subsets have one dimension less; see also Table 2. Thenotation is such that the subscript of B ⋆ denotes the non-zero variables, e.g. B N denotes thetype II subset with N > N = 0 and N = 0.Each set of the Lie contraction hierarchy is the union of an invariant vacuum subset, i.e.,Ω = 0, and an invariant fluid subset, i.e., Ω >
0. To refer to a vacuum [fluid] subset of aBianchi set B ⋆ we use the notation B vac .⋆ [ B fl .⋆ ]. In this spirit, e.g., the type II subset decomposesas B N = B vac .N ∪ B fl .N .In the following we analyze the boundary subsets to the extent needed in order to understandthe asymptotic type IX dynamics. The Bianchi type I subset
The Bianchi type I subset is given by N = 0, N = 0, N = 0 and Ω = 1 − Σ ≥
0; since N , N , N vanish, we denote this subset by B ∅ , cf. Figure 1. The vacuum subset consists of acircle of fixed points— the Kasner circle K , which is characterized by Σ = 1. It is commonto represent different points on K in terms of the Kasner exponents p α ,(Σ , Σ , Σ ) = (3 p − , p − , p −
1) ; p + p + p = 1 , p + p + p = 1 ; (15)each fixed point on K represents a Kasner solution (Kasner metric) with the correspondingexponents. The Kasner circle is divided into six equivalent sectors, denoted by permutationsof the triple (123), where sector ( αβγ ) is characterized by p α < p β < p γ , see Figure 2. Theboundaries of the sectors are six special points that are associated with solutions that are locallyrotationally symmetric (LRS): Q α are given by (Σ α , Σ β , Σ γ ) = ( − , ,
1) or ( p α , p β , p γ ) =( − , , ) and yield the three equivalent LRS solutions whose intrinsic geometry is non-flat;the Taub points T α are given by (Σ α , Σ β , Σ γ ) = (2 , − , −
1) or ( p α , p β , p γ ) = (1 , ,
0) andcorrespond to the flat LRS solutions—the Taub representation of Minkowski spacetime.The quantity p p p (or, equivalently, Σ Σ Σ = 2 + 27 p p p ) is invariant under changes ofthe axes and thus naturally captures the ‘physical essence’ of a solution independent of the THE BIANCHI TYPE IX LIE CONTRACTION HIERARCHY Σ Σ (123)(213)(231)(321) (312) (132)T T T Q Q Q M M M StartFigure 2: The Kasner circle K of fixed points divided into its six equivalent sectors and theLRS fixed points T α and Q α . Sector ( αβγ ) is defined by Σ α < Σ β < Σ γ . The sectors arerelated to each other by permutations of the spatial axes.chosen frame; however, it is typically replaced by the Kasner parameter u through p p p = − u (1 + u ) (1 + u + u ) , where u ∈ [1 , ∞ ] . (16)The Kasner parameter u parameterizes the Kasner exponents uniquely (up to the permutationsymmetry); we have p α = − u u + u , p β = 1 + u u + u , p γ = u (1 + u )1 + u + u , (17)for sector ( αβγ ) of K , where u ∈ (1 , ∞ ). Therefore, each point on sector ( αβγ ) is representedby a unique value of u ∈ (1 , ∞ ). At the boundary points of sector ( αβγ ), which are Q α and T γ ,the Kasner parameter is u = 1 and u = ∞ , respectively. Permuting ( αβγ ) yields a physicallyequivalent state on a different sector; accordingly, each u ∈ (1 , ∞ ) represents an equivalenceclass of six points on K . In contrast, u = 1 describes the three points { Q , Q , Q } ; u = ∞ yields { T , T , T } .While the Bianchi type I vacuum subset coincides with the Kasner circle K , which is givenby Σ = 1, the Bianchi type I perfect fluid subset is the set 1 − Ω = Σ <
1. From (14) it isstraightforward to deduce that there exists a central fixed point, the
Friedmann fixed point
F,given by Σ α = 0 ∀ α , which corresponds to the isotropic Friedmann-Robertson-Walker (FRW)solution. Solutions with 0 < Σ < andending at F. These results rely on the assumption w <
1, see (1).
The Bianchi type II subset
Let us consider the Bianchi type II subset B N γ given by N α = N β = 0, N γ >
0. In thiscase, the γ -direction is singled out, while there exists a discrete symmetry associated with theinterchange of the α - and β -direction. The LRS subset Σ α = Σ β is a subset of codimension onewhich divides the state space into two equivalent parts, the subsets { Σ α > Σ β } and { Σ α < Σ β } (related by permuting the α - and β -axes). THE BIANCHI TYPE IX LIE CONTRACTION HIERARCHY τ − according to τ − = − τ (18)which we do in the remainder of this section; accordingly, approach to the past singularitymeans τ − → ∞ . (In Section 5 we will see that vacuum type II orbits are the building blocksfor the Mixmaster map and the closely related Kasner map (BKL map). It is a well establishedconvention that forward iterations of these maps are directed towards the singularity. To agreewith this convention the use of a past-directed time variable in a discussion of Bianchi type IImodels thus suggests itself.)On B N γ , the Gauss constraint Σ + N γ +Ω = 1 can be used to replace N γ by Ω as a dependentvariable. The system (7) thus becomes d Σ α/β dτ − = (2 − q )Σ α/β + S α/β , d Σ γ dτ − = (2 − q )Σ γ + S γ , d Ω dτ − = − Ω [2 q − (1 + 3 w )] , (19)where q = 2Σ + (1 + 3 w )Ω and S α/β = − − Σ − Ω), S γ = 8(1 − Σ − Ω); we haveΣ + Ω < B vac .N γ , i.e., Ω = 0. There do not exist any fixed pointsin the type II vacuum subset B vac .N γ , but the boundary of the vacuum subset coincides withthe Kasner circle K . The orbits of (19) form a family of straight lines in B vac .N γ , where eachorbit connects one fixed point on K with another fixed point on K ; hence each orbit isheteroclinic [10]. Following the nomenclature of [30] we call these heteroclinic orbits Bianchitype II transitions , because each orbit can be viewed as representing a transition from oneKasner state to another. We denote these transitions by T N γ , where each T N γ emanates from( γαβ ) ∪ Q γ ∪ ( γβα ). If the initial point is a point of sector ( γαβ ), then the final point is a pointof ( αγβ ) ∪ { Q α } ∪ ( αβγ ); interchanging α and β yields the transitions emanating from ( γβα );if the initial point is Q γ , the final point is T γ ; the points T α and T β are not connected withany other fixed point (they are ‘fixed points’ under the present ‘type II map’), see Figure 3.Let (Σ i α , Σ i β , Σ i γ ) = (3 p i α − , p i β − , p i γ −
1) denote the initial fixed point on K ; this point canbe represented in terms of the Kasner parameter u = u i by using (17). The orbit (transition)emanating from this fixed point is given in terms of an auxiliary function η = η ( τ − ) byΣ α/β = 2[1 − η ] + η Σ i α/β , Σ γ = − − η ] + η Σ i γ , (20a)where η is determined by the equation dηdτ − = 2(1 − Σ ) η with (1 − Σ ) = 3 g ( g − η )( η −
1) and g = 1 + u + u − u + u (20b)and the conditions that lim τ − →−∞ η = 1 and lim τ − → + ∞ η = g . The quantity g is in the interval(1 ,
3) for u ∈ (1 , ∞ ); u = 1 corresponds to g = 3 and describes the orbit Q γ → T γ ; u = ∞ corresponds to g = 1 and describes the ‘isolated’ points T α , T β . Since η increases from 1 to g as τ − goes from −∞ to + ∞ , g is called the growth factor [30]. In Section 5, the transitions T N γ , as represented by (20), will appear as the building blocks for the Mixmaster/Kasner map.While there do not exist any fixed points in the vacuum subset of B N γ , there exists one fixedpoint in B N γ with Ω >
0, the
Collins-Stewart fixed point CS γ , which corresponds to one repre-sentation of the LRS solutions found by Collins and Stewart [38]. CS γ is given by (Σ α , Σ β , Σ γ ) = (1 + 3 w )(1 , , −
2) and Ω = 1 − (1 + 3 w ) (which yields N γ = √ − w √ w ). The fixedpoint CS γ is the source (w.r.t. τ − ) for all orbits in B N γ with Ω >
0. In the limit τ − → ∞ allsolutions in B N γ \{ CS γ } converge to fixed points on the Bianchi type I boundary of B N γ : There ASYMPTOTIC SELF-SIMILARITY Σ Σ T T T Q Q Q M Figure 3: The type II transitions T N on the B N subset; by definition, N = 0 along T N ,while N = N = 0. The projections of these transition onto (Σ , Σ , Σ )-space are straightlines, which possess a common focal point M characterized by (Σ , Σ , Σ ) = ( − , , T N , T N on the subsets B N , B N are obtained by permutations of the axes, seeFigure 5. The arrows indicate the direction of time towards the past.exists one orbit (which corresponds to an LRS solution) that converges to F as τ − → ∞ ; everyother orbit converges to a fixed point on ( αγβ ) ∪ { Q α } ∪ ( αβγ ) or ( βγα ) ∪ { Q β } ∪ ( βαγ ) onK , or to T γ (in the LRS case). For a detailed discussion of these results see [10]. The Bianchi type VII subset In anticipation of Theorem 6.1 which implies that generic orbits of Bianchi type IX do nothave α -limit points (w.r.t. the standard future directed time variable τ ) on any of the Bianchitype VII subsets B N N , B N N , B N N , we refrain from giving a detailed discussion of thesesubspaces here. (However, note that in order to prove Theorem 6.1, a detailed understandingof solutions of Bianchi type VII is essential; in fact, in the proof of Theorem 6.1 Bianchitype VII is ubiquitous; we refer to [26] and [27].) In the present context it suffices to note thaton each subset B N α N β there exists a line of fixed points TL γ given by (Σ α , Σ β , Σ γ ) = ( − , − , N α = N β (so that Ω = 0). Since TL γ emanates from the point T γ ∈ K we call it the‘Taub line.’ Like T γ itself, each of fixed points on TL γ is associated with a representation ofMinkowski spacetime (in an LRS type VII symmetry foliation). In the previous section we have given the fixed points associated with the system (7) on B IX . Alocal dynamical systems analysis of the fixed points shows whether or not these points attracttype IX orbits in the limit τ → −∞ . We merely state the results here and refer to [27] fordetails. For a dynamical system on a state space X , the α -limit set α ( x ) of a point x ∈ X is defined as the set of allaccumulation points towards the past (i.e., as τ → −∞ ) of the orbit γ ( τ ) through x . The simplest examples of α -limit sets are fixed points and periodic orbits. In this section we adapt to [10, 26] and use the normal future-directed time variable τ . ASYMPTOTIC SELF-SIMILARITY
Each fixed point K on K \{ T , T , T } is a transversally hyperbolic saddle (one stable mode, see Figure 4, and three unstable modes; the stable manifold of K coincides with avacuum type II transition orbit, see Figure 3). The Taub points { T , T , T } are centersaddles with a two-dimensional unstable manifold and a three-dimensional center mani-fold, where T α is excluded as an α -limit set on the center manifold, see [27]; consequentlythere do not exist any type IX solutions that converge to any of the points on K as τ → −∞ .F The fixed point F on B fl . ∅ is a hyperbolic saddle (with B fl . ∅ as a two-dimensional stablemanifold and an additional three-dimensional unstable manifold). Accordingly, F attractsa two-parametric family of type IX orbits as τ → −∞ . These solutions have a so-calledisotropic singularity.CS α The fixed points CS α ( α = 1 , ,
3) on B fl .N α are hyperbolic saddles (with B fl .N α as a three-dimensional stable manifold and an additional two-dimensional unstable manifold). Theunstable modes are associated with the equations N − β N ′ β | CS α = (1 + 3 w ) (for β = α ).Therefore, each of the fixed points CS α attracts an (equivalent) one-parameter set oftype IX orbits in the limit τ → −∞ .TL α Each fixed point on TL α on B vac .N β N γ is a center saddle. On the three-dimensional centermanifold (which coincides with B vac .N β N γ , α = β = γ = α ) the point acts as a (non-hyperbolic) sink, see [27]. Since there is a two-dimensional unstable manifold, there exists,for each fixed point on TL α , a one-parameter family of type IX orbits that converges to itas τ → −∞ ; these orbits correspond to LRS solutions. (Conversely, generic LRS type IXsolutions converge to TL α , see, e.g.,[10].)PSfrag replacementsΣ Σ Σ N N N N N N T T T Q Q Q M M M StartFigure 4: This figure depicts the Kasner circle K and the stable variables for each sector; N α isthe stable variable in sectors ( αβγ ), ( αγβ ), and at the point Q α . Expressed in the time variable τ − , which is directed towards the past, these variables are the unstable modes. At a given fixedpoint, the associated unstable manifold orbit is a Bianchi type II transition, see Figure 3.Collecting the results we see that the solutions whose α -limit is one of the fixed points form asubfamily of measure zero of the (four-parameter) family of Bianchi type IX solutions. Following When we reverse the direction of time, i.e., when we use τ − instead of τ , we must replace ‘stable’ by‘unstable’. FACTS ABOUT THE MIXMASTER AND KASNER MAP non-generic solutions of Bianchitype IX. Alternatively, to capture the asymptotic behavior of these solution, we use the term pastasymptotically self-similar solutions. (Since a fixed point in the Hubble-normalized dynamicalsystems formulation corresponds to a self-similar solution, see e.g. [10], solutions that convergeto a fixed point are asymptotically self-similar.)The past asymptotically self-similar solutions comprise the LRS Bianchi type IX solutions. Asseen above, generic LRS solutions converge to TL α towards the past (and each solution thatconverges to TL α is LRS), but there exist exceptional LRS solutions that converge to F or CS α .The remaining orbits whose limit point is either F or CS α correspond to past asymptoticallyself-similar solutions that are non-LRS. Clearly, every solution that converges to F or CS α is anon-vacuum solution, since Ω = 0 at F and CS α .It is natural to ask how the non-generic orbits are embedded in the state space B IX . The LRSorbits form the three LRS subsets LRS α , which are the hyperplanes given by the conditionsΣ β = Σ γ , N β = N γ , where ( αβγ ) ∈ { (123) , (231) , (312) } . The orbits whose α -limit set is thefixed point CS α (for some α ) form the set CS α in B IX ; we call CS α the Collins-Stewart manifold.The local analysis of the fixed point CS α and the regularity of the dynamical system (7) implythat the Collins-Stewart manifold CS α is a two-dimensional surface; it can be viewed as a two-dimensional manifold with boundary embedded in B IX (where this boundary corresponds toan orbit in B VII ). Analogously, the orbits whose α -limit set is the fixed point F form the set F in B IX , which we call the isotropic singularity manifold, since solutions converging to F arethose with an isotropic singularity. The isotropic singularity manifold F is a three-dimensionalhypersurface; it can be viewed as a three-dimensional manifold with boundary.Generic Bianchi type IX models are those that are not asymptotic self-similar and thus con-stitute examples for asymptotic self-similarity breaking; for other such examples, see [39, 40].The central theme in this paper is the past asymptotic behavior of the generic models. The
Mixmaster attractor A IX (alternatively referred to as the Bianchi type IX attractor) isdefined to be the subset of B IX given by the union of the Bianchi type I and II vacuumsubsets, i.e., A IX = B vac . I ∪ B vac . II . Since the type II vacuum subset consists of three equivalentrepresentations we obtain A IX = K ∪ B vac .N ∪ B vac .N ∪ B vac .N . (21)In this section we investigate the structures that the flow of dynamical system (7) induces on A IX . In particular we discuss the Mixmaster map, the Kasner map, and the era map. To agreewith the well-established convention for these maps, the direction of time will be taken towardsthe past. The Mixmaster, Kasner, and era maps
In Section 3 we have seen that the vacuum type II orbits, i.e., the orbits on B vac .N α , α = 1 , , ,see Figures 3 and 5. In accord with Section 3 we refer to these orbits as transitions and denotethem by T N α , α = 1 , , sequence of transitions , also known as a heteroclinicchain . Since each fixed point on K (except for the Taub points) is the initial value for one FACTS ABOUT THE MIXMASTER AND KASNER MAP T T T Q Q Q PSfrag replacements T T T Q Q Q PSfrag replacements T T T Q Q Q Figure 5: Projection of the type II transitions T N T N , T N on the type II subsets B vac .N , B vac .N , B vac .N onto (Σ , Σ , Σ )-space. Concatenation of these transition orbits yields sequencesof transitions. The arrows indicate the direction of time towards the past.single transition, see Figures 4 and 5, the concatenation of transitions is unique: Each fixedpoint (except for the Taub points) generates a unique sequence of transitions. Note, however,that the ‘direction of time’ is relevant. For each fixed point P on K , which is not one of theTaub points, there exists one single transition emanating from P at τ − = −∞ , but there are twotransitions converging to P as τ − → ∞ . Therefore, concatenating transitions in the reverseddirection of time leads to ambiguities. (In terms of the standard future-directed time variable τ we have the converse statement: It is possible to make unambiguous retrodictions, but notpredictions.)Let l = 0 , , , . . . and let P l ∈ K denote the initial point of the l th transition (P l is also theend point of the ( l − th transition). We refer to the sequence (P l ) l ∈ N of Kasner fixed points,which is induced by the sequence of transitions, as being generated by the Mixmaster map . TheMixmaster map can be visualized by a map in (Σ , Σ , Σ )-space, obtained by inscribing K in a triangle with corners at (Σ , Σ , Σ ) = ( − , ,
2) and cyclic permutations, from which the(projections of the) transition orbits ‘originate’ as straight lines; see Figure 6.PSfrag replacementsΣ Σ Σ T T T Q Q Q M M M StartFigure 6: Concatenating type II transition orbits we obtain sequences of transitions—heteroclinic chains. The discrete map governing the associated sequence of fixed points onK is the Mixmaster map. The arrows indicate the direction of time towards the past.Let the initial Kasner state of a transition be represented by the Kasner parameter u = u i ,where we assume u i < ∞ , since neither of the Taub points T , T , T can be the initial valuefor a transition. Inserting (15) and (17) into (20) we find that a transition maps the Kasner FACTS ABOUT THE MIXMASTER AND KASNER MAP u i to the parameter u f , where u f = ( u i − u i ∈ [2 , ∞ ) , ( u i − − if u i ∈ [1 , . (22)The information contained in this Kasner map suffices to represent the collection of all transitionorbits (as a whole). However, for each particular mapping u i u f , there exist six (equivalent)associated transitions; this is simply because u i characterizes the initial Kasner fixed point onK only up to permutations of the axes. Hence, in order to reconstruct a particular transitionfrom (22), we supplement (22) with information about the initial sector of the transition, whichdetermines the position of the axes.In terms of the Kasner parameter, a sequence of transitions corresponds to an iteration of (22).Let l = 0 , , , . . . and let u l denote the initial Kasner state of the l th transition. This transitionmaps u l to u l +1 , i.e., u l l th transition −−−−−−−−−→ u l +1 : u l +1 = ( u l − u l ∈ [2 , ∞ ) , ( u l − − if u l ∈ [1 , . (23)We refer to this map as the (iterated) Kasner map (which is also known as the BKL map [2]).Since each value of the Kasner parameter u ∈ (1 , ∞ ) represents an equivalence class of sixKasner fixed points, the Kasner map can be regarded as the map induced by the Mixmastermap on these equivalence classes via the equivalence relation.In a sequence ( u l ) l =0 , , ,... that is generated by the Kasner map (23), each Kasner state u l is called an epoch . Every sequence ( u l ) l =0 , , ,... possess a natural partition into pieces (whichcontain a finite number of epochs each) where the Kasner parameter is monotonically decreasingaccording to the simple rule u l u l +1 = u l −
1; these pieces are called eras [2]. An era beginswith a maximal value u l in of the Kasner parameter (where u l in is generated from u l in − by u l in =[ u l in − − − ), continues with a sequence of Kasner parameters obtained via u l u l +1 = u l − u l out that satisfies 1 < u l out <
2, so that u l out +1 = [ u l out − − begins a new era.6 . → . → . → . → . → . | {z } era → . → . → . | {z } era → . → . | {z } era → . → . . . | {z } era (24)Let us denote the initial (= maximal) value of the Kasner parameter u in era number s (where s = 0 , , , . . . ) by u s . Following [2, 7] we decompose u s into its integer part k s = [ u s ] and itsfractional part x s = { u s } , i.e., u s = k s + x s , where k s = [ u s ] , x s = { u s } . (25)The number k s represents the (discrete) length of era s , which is simply the number of Kasnerepochs it contains. The final (= minimal) value of the Kasner parameter in era s is given by1 + x s , which implies that era number ( s + 1) begins with u s +1 = 1 x s = 1 { u s } . The map u s u s +1 is (a variant of) the so-called era map ; starting from u = u it recursivelydetermines u s , s = 0 , , , . . . , and thereby the complete Kasner sequence ( u l ) l =0 , ,... . In the exceptional case u i = 1 there exist only three (equivalent) associated transitions: Q → T , Q → T ,Q → T . FACTS ABOUT THE MIXMASTER AND KASNER MAP u = k + 1 k + 1 k + · · · = [ k ; k , k , k , . . . ] . (26)The fractional part of u is x = [0; k , k , k , . . . ]; since u is the reciprocal of x we find u = [ k ; k , k , k , . . . ] . (27)Therefore, the era map is simply a shift to the left in the continued fraction expansion, u s = [ k s ; k s +1 , k s +2 , . . . ] u s +1 = [ k s +1 ; k s +2 , k s +3 , . . . ] . (28)The properties of the Kasner sequence depend on the initial value u = u .(i) If and only if the initial Kasner parameter u is a rational number, i.e., if and only if u ∈ Q , then its continued fraction representation is finite, i.e., u = [ k ; k , k , . . . , k n ] , (29i)where k n >
1. Therefore, there exists only a finite number of eras (where the last onebegins with u n = k n ), and the Kasner sequence is finite. At the end of era number n ,the Kasner parameter reaches u = 1, which subsequently terminates the recursion (23).Since Q is a set of measure zero in R , this case is non-generic.(ii) A quadratic irrational (quadratic surd) is an algebraic number of degree 2, i.e., an irra-tional solution of a quadratic equation with integer coefficients. If and only if the initialKasner parameter u is a quadratic irrational, i.e., if and only if u = q + √ q , where q ∈ Q and q ∈ Q is not a perfect square (i.e., √ q Q ), then its continued fractionrepresentation is periodic, i.e., u = (cid:2) k ; k , . . . , k n , (¯ k , . . . , ¯ k ¯ n ) (cid:3) ; (29ii)the notation is such that the part in parenthesis, i.e., (¯ k , . . . , ¯ k ¯ n ), is repeated ad infini-tum. Consequently, the era map becomes periodic (after the n th era), and we thus obtaina periodic sequence of eras and a periodic Kasner sequence ( u l ) l =0 , , ,... . It is straight-forward to see that while the period of the era sequence is ¯ n , the period of the Kasnersequence is (¯ k + · · · + ¯ k ¯ n ); see the examples below. Since the set of algebraic numbers ofdegree two (or equivalently the set of equations with integer coefficients—it is a subset of N ) is a countable set, case (ii) is also non-generic.(iii) An irrational number is called badly approximable if its Markov constant is finite. If andonly if u is badly approximable, then the coefficients (partial quotients) in its continuedfraction representation are bounded, i.e., u = (cid:2) k ; k , k , k , . . . (cid:3) with k i ≤ K ∀ i (29iii)for some positive constant K . Consequently, the sequence of eras and the Kasner sequence( u l ) l =0 , , ,... are bounded, i.e., u l ≤ K ∀ l . Obviously, case (ii) is a subcase of case (iii). If u = q + √ q > q − √ q ∈ ( − , u = ˆ (¯ k , . . . , ¯ k ¯ n ) ˜ , i.e.,the continued fraction is purely periodic without any preperiod. For x ∈ R , let k x k denote the distance from x to the nearest integer, i.e., k x k = min n ∈ Z | x − n | . The Markovconstant M ( x ) of a number x ∈ R \ Q is defined as M ( x ) − = lim inf N ∋ n →∞ n k nx k , see [41]. It is known that M ( x ) ≥ √ x ∈ R \ Q . FACTS ABOUT THE MIXMASTER AND KASNER MAP u is a well approximable irrational number,then the partial quotients k i in the continued fraction representation u = (cid:2) k ; k , k , k , . . . (cid:3) (29iv)are unbounded (and we can construct a diverging subsequence from the sequence of partialquotients ( k i ) i ∈ N ). This is the generic case, and hence generically the Kasner sequence( u l ) l =0 , , ,... is infinite and unbounded.In terms of continued fractions, the Kasner sequence ( u l ) l ∈ R generated by u = (cid:2) k ; k , k , . . . (cid:3) is u = u = (cid:2) k ; k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → . . . → (cid:2) k , k , . . . (cid:3) → u = (cid:2) k ; k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → . . . → (cid:2) k , k , . . . (cid:3) → u = (cid:2) k ; k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → (cid:2) k − k , k , . . . (cid:3) → . . . Let us give some examples for periodic era sequences and
Kasner sequences . If u = [(1)] =(1 + √ /
2, which is the golden ratio, then u s = (1 + √ / ∀ s . It follows that the Kasnersequence is also a sequence with period 1,( u l ) l ∈ N : (cid:0) √ (cid:1) → (cid:0) √ (cid:1) → (cid:0) √ (cid:1) → (cid:0) √ (cid:1) → (cid:0) √ (cid:1) → . . . If u = [(2)] = 1 + √
2, then u s = 1 + √ ∀ s ; hence the era sequence is a sequence of period 1.However, the associated Kasner sequence has period 2,( u l ) l ∈ N : (1 + √ → √ | {z } era → (1 + √ → √ | {z } era → (1 + √ → √ | {z } era → (1 + √ → √ | {z } era → . . . Analogously, the initial value u = [(3)] = (3+ √ / u = [(2 , p / ≃ . u n = [(2 , ≃ . n and u n = [(4 , ≃ . n . The associated Kasner sequence ( u l ) l ∈ N has period 6,2 . → . | {z } era → . → . → . → . | {z } era → . → . | {z } era → . → . → . → . | {z } era → . . . In the state space description of sequences (in terms of the Mixmaster map), an epoch is simplya point P l on the Kasner circle. (It is one of the six points in the equivalence class associatedwith the Kasner parameter u l .) Transitions connect epochs and thus generate the Mixmastermap.The Kasner parameter u can be employed to measure the (angular) distance of a point P onK from the Taub points or from the non-flat LRS points: If u ≫
1, then P is at an angulardistance of approximately u − from one of the Taub points. On the other hand, in the vicinityof the non-flat LRS points (where u − ≪ u − u <
2, then P is closer to one of the non-flatLRS points Q α than to any of the Taub points T α . Therefore, in the state space picture, an era can be described as a (finite) sequence of points K ∋ P l , l in ≤ l ≤ l out , obtained from FACTS ABOUT THE MIXMASTER AND KASNER MAP l in that is close to one of the Taub points T α (the precedingpoint P l in − was closer to one of the other two Taub points T β , β = α , than to T α ); thenthe distance from the Taub point T α monotonically increases until, for P l out , it exceeds thedistance to the non-flat LRS points; this is the terminal point for the era; it is connected bythe following transition with the initial point of the next era; a good illustration is Figure 7(d),where each era consists of three epochs. Note that due to the equivalence of Kasner points,there exist several realizations of one and the same Kasner sequence as Mixmaster sequencesin the state space picture; this is exemplified by the heteroclinic cycles in Figure 7(a) and 7(b).The state space description of the cases (i)–(iv), which are characterized by the initial Kasnerparameter u = u , is the following:(i) Iff u ∈ Q , see (29i), the Mixmaster sequence of Kasner fixed points (P l ) l =0 , ,... is finite.After a finite number of transitions, at the end of era n , the sequence reaches one of theLRS points Q α (where u = 1); a last transition follows, namely the transition Q α → T α ,and the sequence terminates in one of the Taub points.(ii) Iff u is a quadratic surd, i.e., u = q + √ q for some q , q ∈ Q where q is not a perfectsquare, see (29ii), then the Mixmaster sequence of Kasner points (P l ) l ∈ N is eventuallyperiodic, where the period is a multiple of (¯ k + · · · + ¯ k ¯ n ); see Figure 7. Viewed as aperiodic sequence of transitions (which are heteroclinic orbits) we obtain a heterocliniccycle. In Figure 7 we give some of the heteroclinic cycles associated with Kasner sequenceswith periods 1, 2, and 3 in their projection onto (Σ , Σ , Σ )-space. Note that due topermutation symmetry there are several cycles associated with a given periodic Kasnersequence.(iii) Iff u is a badly approximable irrational number, see (29iii), there exists a neighborhoodof the Taub points T α such that the Mixmaster sequence of Kasner points (P l ) l ∈ N doesnot enter this neighborhood. This is simply because there exists a maximal value of thesequence ( u l ) l ∈ N .(iv) Iff u is a well approximable irrational number, see (29iv), then the Mixmaster sequence(P l ) l ∈ N comes arbitrarily close to the Taub points. This is the generic case.In the following we analyze the generic case (iv) in more detail. Let u be a well approximableirrational number, i.e., a number whose continued fraction expansion u = (cid:2) k ; k , k , k , . . . (cid:3) (30)defines an unbounded sequence ( k i ) i ∈ N . By construction, era number i contains k i epochs(which we call its length). A natural question to ask concerns the distribution of the partialquotients k i . For a ‘typical’ well approximable irrational number, how often does the number1 appear in the sequence ( k i ) i ∈ N ? How often the number 2? And what about the number1000? The answer is given by Khinchin’s law [42]. Let P n ( k = m ) denote the probability thata randomly chosen partial quotient among ( k , . . . , k n ) equals m ∈ N . In the asymptotic limit,i.e., for P ( k = m ) = lim n →∞ P n ( k = m ) we have P ( k = m ) = log (cid:0) m + 1 m + 2 (cid:1) − log (cid:0) mm + 1 (cid:1) , (31)i.e., the partial quotients of the continued fraction representation of (30) are distributed likea random variable whose probability distribution is given by (31). (By (31), the number 1appears in 42% of the slots, the number 2 in 17%, and the number 1000 in 1 . ∗ − % of theslots.) Khinchin’s law applies for almost all numbers u . FACTS ABOUT THE MIXMASTER AND KASNER MAP Σ Σ T T T Q Q Q (a) One of the two heteroclinic cyclesassociated with the Kasner sequencegenerated by u = [(1)]. PSfrag replacementsΣ Σ Σ T T T Q Q Q (b) The second of the two heterocliniccycles associated with the Kasner se-quence generated by u = [(1)]. PSfrag replacementsΣ Σ Σ T T T Q Q Q (c) One of the heteroclinic cycles asso-ciated with the Kasner sequence gener-ated by u = [(2)]. PSfrag replacementsΣ Σ Σ T T T Q Q Q (d) One of the heteroclinic cycles asso-ciated with the Kasner sequence gener-ated by u = [(3)]. Figure 7: Examples of heteroclinic cycles associated with era sequences with period 1. TheKasner sequences have period 1, 2, and 3, respectively; the period of the heteroclinic cycles isa multiple of that period. Note that the direction of time is towards the past.For a (generic) Kasner sequence ( u l ) l ∈ N with initial parameter u = [ k ; k , k , . . . ] and itsassociated era sequence ( u s ) s ∈ N , where u s = [ k s ; k s +1 , k s +2 , . . . ], the expression P ( k = m ) of (31)represents the probability that a randomly chosen era of ( u s ) s ∈ N has length m ; this correspondsto the probability that the initial value of an era is contained in the interval [ m, m + 1). In thismanner, the probability distribution (31) makes possible a stochastic interpretation of genericKasner sequences.The probability distribution (31) results in extraordinary properties of the Mixmaster/Kasnermap, which will be of crucial importance in the considerations of Section 7. We do not discussdetails here but refer to future work; however, we cannot refrain from giving a teaser: For ageneric Kasner sequence ( u l ) l ∈ N and its associated era sequence ( u s ) s ∈ N there exist infinitelymany eras such that the length (i.e., the number of epochs) of the n th era is larger than n log n ;however, for sufficiently large n , the length is guaranteed to be bounded by n log n . (For aproof of this result by Borel and Bernstein see [43]; see also [44].) Properties of this kindunderline the remarkable intricacies of the heteroclinic structures on the Mixmaster attractor. MIXMASTER FACTS In this section we turn to what is known about the past asymptotic dynamics of generic type IXsolutions. The main Mixmaster fact is Ringstr¨om’s ‘Bianchi type IX attractor theorem’.
The main Mixmaster fact
Consider a solution of Bianchi type IX that is either vacuum or associated with a perfectfluid satisfying − < w <
1. Recall that such a solution is called generic if it is not pastasymptotically self-similar, i.e., if its α -limit set is neither the point F, nor any of the pointsCS α , nor a point on TL α ; in other words, a generic solution corresponds to an orbit in B IX that is neither contained in F , nor in CS α , nor in LRS α . Therefore, the set of generic Bianchitype IX states is an open set in B IX . (To conform with [10, 26] we use the future directed timevariable τ .)The main results concerning generic Bianchi type IX models are due to Ringstr¨om [26]; theseresults rest on earlier work that is reviewed and derived in [10], and on [24, 25]. In the followingwe state the main theorem in a version adapted to our purposes. Theorem 6.1 ([26]) . Let (Σ , Σ , Σ , N , N , N )( τ ) be a generic solution of Bianchi type IX,i.e., a generic solution of (7) in B IX . Then ∆ II = N N + N N + N N → and Ω → as τ → −∞ . Note that this Theorem applies to both the fluid and the vacuum case; in the latter case (32)becomes ∆ II → ≡ α -limitset of a generic Bianchi type IX solution is non-empty and must contain a point on the Kasnercircle. The main part of the proof deals with the fact that the function ∆ II ( τ ) is in generalnot monotone. There exist times where ∆ II ( τ ) increases (as τ → −∞ ); the associated growthmust therefore be controlled and shown to be negligible compared to the overall decrease in∆ II . This is done by a careful analysis of the equations and (approximate) solutions. In [27]we give an alternative and relatively short and succinct proof of Theorem 6.1 which is basedon an in-depth understanding of the hierarchical structure of the dynamical system (7) (asrepresented by Figure 1). It is important to note, however, that either of the proofs fail inthe other Bianchi types that are conjectured to exhibit an oscillatory approach towards thesingularity, i.e., the proofs fail for types VI − / and VIII. This is unfortunate, since there arereasons to believe that these models are more relevant than type IX as regards the dynamicsof generic (inhomogeneous) cosmologies, see [27].Using the concept of the Mixmaster attractor, cf. (21), we obtain an equivalent formulation ofTheorem 6.1: Let X ( τ ) = (Σ , Σ , Σ , N , N , N )( τ ) be a generic solution of Bianchi type IX.Then k X ( τ ) − A IX k → τ → −∞ ) , (33)where the distance k X − A IX k is given as min Y ∈A IX k X − Y k .Theorem 6.1 thus states that the attractor of generic type IX solutions resides on A IX ; how-ever, whether the past attractor is in fact A IX or merely a subset thereof remains open. (Theterminology ‘Mixmaster attractor’ is seductive but might turn out to be quite misleading.) Like-wise, the theorem does not provide any direct information about the details of the asymptoticbehavior of solutions. MIXMASTER FACTS Consequences
The prerequisite for a deeper understanding of the asymptotic behavior and the oscillatorydynamics of generic type IX solutions is an understanding of the Mixmaster attractor. InSection 5 we have identified the structures on the Mixmaster attractor A IX that are induced bythe flow of the dynamical system: heteroclinic cycles [case (ii)] and finite [case (i)] and infiniteheteroclinic sequences [cases (iii) and (iv)]. All these structures qualify (a priori) as possible α -limit sets of generic type IX orbits. In conjunction with these results, the main theorem 6.1implies a number of further facts about the attractor—‘Mixmaster attractor facts’, which wegive as a list of corollaries . (For proofs see [27], and also [26].)1. A generic, i.e. not past asymptotically self-similar, type IX orbit possesses an α -limitpoint on A IX .2. If P ∈ A IX is an α -limit point of a type IX orbit, then the entire heteroclinic cycle/sequence(Mixmaster sequence) through P must be contained in the α -limit set.3. If one of the Taub points { T , T , T } is an α -limit point of a type IX orbit, then the α -limit set contains Kasner fixed points associated with arbitrarily large values of theKasner parameter u .4. For generic solutions of Bianchi type IX the Weyl curvature scalar C abcd C abcd (and there-fore also the Kretschmann scalar) becomes unbounded towards the past.5. Taking into account both the expanding and contracting phases of Bianchi type IX so-lutions, generic Bianchi type IX initial data generate an inextendible maximally globallyhyperbolic development associated with past and future singularities where the curvaturebecomes unbounded.6. Convergence to the Mixmaster attractor is uniform on compact sets of generic initial data:Let X be a compact set in B IX that does not intersect any of the manifolds F , CS α , LRS α ,so that each initial data ˚ x ∈ X generates a generic type IX solution. Let X (˚ x ; τ ) denotethe type IX solution with X (˚ x,
0) = ˚ x . Then k X (˚ x ; τ ) − A IX k → τ → −∞ ) (34)uniformly in ˚ x ∈ X .Corollary 3 implies that the α -limit set contains an infinite set of Kasner fixed points in aneighborhood of the Taub point(s), but this set is not necessarily a continuum of fixed points;cf. the previous discussion about possible α -limit sets. If a Kasner point with u ∈ Q is containedin the α -limit of an orbit, so is a Taub point. This is an immediate consequences of the resultsof Section 5, case (i). On the other hand, if the α -limit set of an orbit is a heteroclinic cycleor a heteroclinic sequence (possibly in combination with a cycle) associated with cases (ii) and(iii) of Section 5, then there exists a neighborhood of the Taub points whose intersection withthe α -limit set is empty.Note that the oscillatory behavior of asymptotic type IX dynamics, which we unfortunatelyknow no details about, constitutes an example of asymptotic self-similarity breaking [39]. Inorder to make progress as regards the details of the asymptotic oscillatory behavior, it is naturalto first establish the connection between the Mixmaster/Kasner map and dynamics for a finitetime interval ∆ τ , discussed next. MIXMASTER FACTS Finite Mixmaster shadowing
To make contact between the Mixmaster map and Bianchi type IX asymptotic dynamics weintroduce the concept of finite Mixmaster shadowing which formalizes the following basic idea:Given a sequence of transitions we can choose type IX initial data sufficiently close to the initialdata of the sequence so that the type IX solution generated by this data remains close to thesequence for some ‘time’.Let P be a Kasner fixed point (but P
6∈ { T , T , T } ) and let u be the associated value ofthe Kasner parameter. There exists a unique sequence of transitions ( T l ) l ∈ N and an associatedMixmaster sequence (P l ) l ∈ N with P as initial data; the associated Kasner sequence is ( u l ) l ∈ N .(If u ∈ Q , the sequence terminates at one of the Taub points after a finite number of transi-tions.) We shall make the definition that a type IX solution shadows a finite piece ( T l ) l =0 , ,...,L of the sequence of transitions if it is contained in a prescribed (small) tubular neighborhoodof ( T l ) l =0 , ,...,L . However, a standard ǫ -neighborhood of the sequence fails to be a reasonablemeasure of closeness, because in the vicinity of a Taub point the transitions lie so dense thatconsecutive transitions are not separated from each other by their respective ǫ -neighborhoods.Therefore, the introduction of adapted tubular neighborhoods is necessary to take into accountthe sensitivity of the flow at the Taub points and to capture more accurately the intuitive ideaof shadowing.Let ǫ > Taub-adapted neighborhood ’ of the Mixmaster sequence (P l ) l ∈ N is asequence ( U l ) l ∈ N of open balls, where U l is centered at the point P l and has radius ǫu − l , i.e., U l = { X ∈ B IX : k X − P l k < ǫu − l } . (The radius ǫu − l is chosen to ensure that the intersectionof the ball U l with the Kasner circle induces more or less a standard ǫ -neighborhood ( u l − ǫ, u l + ǫ )of the Kasner parameter u l ; recall from Section 5 that u − measures the angular distance ofa fixed point from a Taub point.) A Taub-adapted tubular neighborhood of the sequence oftransitions ( T l ) l ∈ N is the sequence of tubes ( V l ) l ∈ N that linearly interpolate between U l and U l +1 .Based on this definition we say that a type IX solution X ( τ ) shadows a finite piece ( T l ) l =0 , ,...,L of the sequence of transitions if it moves in a prescribed Taub-adapted tubular neighborhood( T l ) l =0 , ,...,L , i.e., if there exists a sequence of times ( τ l ) l =0 , ,...,L +1 such that X ( τ ) is containedin V l for all τ ∈ ( τ l +1 , τ l ] for all 0 ≤ l ≤ L .Making use of these concepts, a formulation of finite Mixmaster shadowing is the following:Let ǫ > L ∈ N . Consider the sequence of transitions ( T l ) l ∈ N emanating from an initialKasner point P and its Taub-adapted tubular neighborhood (associated with ǫ ). Then thereexists δ ǫ > X ( τ ) that is generated by initial data X with k X − P k < δ ǫ shadows the finite piece ( T l ) l =0 , ,...,L of the sequence of transitions. (A proofof this statement—in a slightly different form—has been given by Rendall [24]. Alternatively,one can invoke the regularity of the dynamical system, the center manifold reduction theoremand continuous dependence on initial data.) Evidently, δ ǫ depends on the choice of ǫ and L .More importantly, however, δ ǫ depends on the position of P —shadowing is not uniform; inparticular, if we consider a series of initial points that approach one of the Taub points, then δ ǫ necessarily converges to zero along this series—shadowing is more delicate in the vicinity ofa Taub point. We will return to this issue in some detail in the next Section.Finite Mixmaster shadowing concerns any generic type IX orbit X ( τ ). Let P ∈ K be an α -limitpoint of the type IX orbit X ( τ ); without loss of generality we may assume that P is not oneof the Taub points. (The existence of such a point is ensured by Corollaries 1–3 of Section 6.)For simplicity we assume that P is associated with an irrational value of the Kasner parameter,which guarantees that the sequence (P l ) l ∈ N emanating from P = P is an infinite sequence.Since P is an α -limit point of X ( τ ), there exists a sequence of times ( τ n ) n ∈ N , τ n → −∞ ( n → ∞ ), such that X ( τ n ) → P ( n → ∞ ). Therefore, we observe a recurrence of phases, wherethe orbit X ( τ ) shadows (P l ) l ∈ N with an increasing degree of accuracy, i.e., shadowing takes MIXMASTER FACTS l ) l ∈ N is finite and terminates at a Taubpoint. X ( τ ) will shadow this finite sequence recurrently with an increasing degree of precision.)We will return to this issue in some more detail in the subsection ‘Stochastic Mixmaster beliefs’of Section 7.The concept of shadowing leads directly to the concept of approximate sequences which weintroduce next. Consider a generic type IX orbit X ( τ ) = (Σ , Σ , Σ , N , N , N )( τ ) and thefunction k X ( τ ) − K k , where the distance k X − K k is given as min Y ∈ K k X − Y k . When theorbit X ( τ ) traverses a (sufficiently small) neighborhood of a fixed point on K \{ T , T , T } ,the function k X ( τ ) − K k exhibits a unique local minimum. This is immediate from thetransversal hyperbolic saddle structure of the fixed point. (However, the flow in the vicinityof the Taub points is more intricate, since these points are not transversally hyperbolic.) Itfollows that the function k X ( τ ) − K k can be used to partition X ( τ ) into a sequence of segmentsin a straightforward manner: The local minima of k X ( τ ) − K k form an infinite sequence( τ l ) l ∈ N such that τ l → −∞ as l → ∞ . (This follows directly from Corollaries 1 and 2 because X ( τ ) has α -limit point(s) on the Kasner circle and α -limit points on the type II subset.) A segment of X ( τ ) is defined to be the solution curve between two consecutive minima, i.e., theimage of the interval ( τ l +1 , τ l ]. (Note that τ → −∞ in the approach to the singularity, whilethe discrete ‘time’ l is past-directed, i.e., l → ∞ towards the singularity. This convention ischosen to agree with the standard convention for the Mixmaster and the Kasner map.) In theasymptotic regime, i.e., in the approach to the Mixmaster attractor, finite shadowing entailsthat a finite sequence of segments will resemble a finite sequence of type II transitions; thisassumes, however, that the type IX orbit does not come too close to any of the Taub points.(In the neighborhood of a Taub point the flow of the dynamical system is much more intricate.This might yield recurring interruptions of the ‘standard behavior’.) We call the type IX orbitin its segmented form an approximate sequence of transitions .In addition, we define a sequence of ‘check points’ that is associated with an approximatesequence of transitions. Each minimum τ l of k X ( τ ) − K k is associated with a Kasner fixedpoint ˇP l (a ‘check point’) that is defined as the minimizer on K of the distance between X ( τ l )and K . (Note that the check points (ˇP l ) l ∈ N do not lie on the type IX orbit X ( τ ).) Sinceeach check point ˇP l is associated with a value ˇ u l of the Kasner parameter, the sequence (ˇP l ) l ∈ N induces a sequence (ˇ u l ) l ∈ N . The value ˇ u l +1 is in general not generated from ˇ u l by the exactKasner map (23), but differs from that value by an error of δ ˇ u l . In our terminology, the sequence(ˇP l ) l ∈ N is an approximate Mixmaster sequence; (ˇ u l ) l ∈ N is an approximate Kasner sequence, seeFigure 8.The approximate Kasner sequence ˇ u l associated with a type IX orbit X ( τ ) does not followthe Kasner map (23) exactly. A natural question to ask, however, is whether the errors δ ˇ u l of the approximate Kasner sequence converge to zero as l → ∞ or not. If δ ˇ u l → l → ∞ for a type IX orbit, this means that its dynamics is completely described by an ‘asymptoticMixmaster/Kasner map’, i.e., by a map that converges to the Mixmaster/Kasner map towardsthe singularity. However, if δ ˇ u l l → ∞ , then the evolution is interrupted repeatedly—infinitely many times—by phases where the dynamics is completely different from the Mixmas-ter dynamics, e.g., ‘eras of small oscillations’ [2]. (In the present state space description an eraof small oscillations is associated with type VII behavior in the vicinity of one of the Taub linesTL α , where N β and N γ are small and of the same order; for details we refer to the discussionof type VII dynamics in [27].) In the subsection ‘Stochastic Mixmaster beliefs’ of Section 7 we The definition of a partition of X ( τ ) into segments seems natural; it is important to note, however, that anydefinition depends on the formulation of the problem and is to a certain extent arbitrary. For instance, instead ofusing the minima of k X ( τ ) − K k one might prefer to analyze the projection of the orbit onto (Σ , Σ , Σ )-spaceand use the extrema of Σ ( τ ). However, the conclusions drawn from any construction of segments are quiteinsensitive to the details of the definition. MIXMASTER BELIEFS X ( τ l ) X ( τ l +1 ) ˇP l ˇP l +1 ˇ δ l X ( τ ) K M i x m a s t e r m a p K a s n e r m a p ( δ ˇ u l ↔ ) δ ˇP l Figure 8: A type IX orbit decomposes into segments X ( τ l ) → X ( τ l +1 ). The points X ( τ l ), l ∈ N , are the local minima of the distance between X ( τ ) and K . The ‘check point’ ˇP l isthe point on K that is closest to X ( τ ) (for τ in a neighborhood of τ l ). Since each checkpoint ˇP l is associated with a value ˇ u l of the Kasner parameter, the ‘approximate Mixmastersequence’ (ˇP l ) l ∈ N induces an ‘approximate Kasner sequence’ (ˇ u l ) l ∈ N . In general, ˇP l +1 /ˇ u l +1 arenot generated from ˇP l /ˇ u l by the exact Mixmaster/Kasner map (23), but differ by an error δ ˇP l / δ ˇ u l .will investigate the behavior of δ ˇ u l along type IX orbits in detail. Attractor beliefs
There remain several important open problems. In the following we will address the mostpressing questions; in particular we will transform vague beliefs into refutable conjectures, andgive arguments in their favor.An immediate question is the following: What are the actual α -limit sets of type IX orbits?Consider a type IX orbit γ . The α -limit set α ( γ ) contains a number of Kasner fixed points (andthe associated type II transitions), each of which is characterized by a particular value of theKasner parameter u . The set of Kasner parameters obtained in this way from α ( γ ) we denoteby U ( γ ). The question of which form U ( γ ) can take for different orbits γ is open, the mostsignificant issues being the following:(i) Is it possible that there exists γ whose limit set U ( γ ) consists of rational numbers only? Itis straightforward to exclude that U ( γ ) coincides with Q itself (or a dense subset thereof),the reason being that α -limit sets are necessarily closed. Another a priori constraint isthat U ( γ ) must contain u = ∞ (which characterizes the Taub points) if U ( γ ) containsa rational number; see Section 5. Corollary 3 of Section 6 then implies that U ( γ ) isunbounded, i.e., U ( γ ) contains arbitrarily large values of the Kasner parameter. A setthat is compatible with these basic requirements is, e.g., U ( γ ) = N ∪ {∞} . Whether thereexist orbits γ such that U ( γ ) takes this (or a related) form remains open.(ii) Is it conceivable that there exist orbits γ such that α ( γ ) is a heteroclinic cycle, see Fig-ure 7? In this case the set U ( γ ) is generated by a quadratic surd u = [(¯ k , ¯ k , . . . , ¯ k n )]via the Kasner map; see Section 5; in particular, U ( γ ) is finite. However, whether orbits γ with this particular past asymptotic behavior really exist is an open problem. MIXMASTER BELIEFS γ such that U ( γ ) is bounded but contains infinitely many u -values?Is the Kasner sequence generated by a badly approximable number a candidate? TheKasner sequence ( u l ) l ∈ N generated by a badly approximable number is an infinite sequencethat is bounded. However, there must be at least one accumulation point of this sequence;if U ( γ ) contains ( u l ) l ∈ N , then U ( γ ) must contain the accumulation point of the sequenceas well (since α -limit sets are closed). If this accumulation point is a well approximablenumber, U ( γ ) cannot be bounded; however, if the accumulation point is a quadratic surd,no inconsistencies arise. Hence, a priori, there might exist orbits γ such that U ( γ ) is notfinite but still bounded. Whether this is indeed the case is doubtful, but hard to excludea priori.An open problem that might be quite separate from the questions raised above concerns thebehavior of all type IX orbits save a set of measure zero. Definition.
The past attractor of a dynamical system given on a state space X is defined asthe smallest closed invariant set A − ⊆ X such that α ( p ) ⊆ A − for all p ∈ X apart from a setof measure zero [45]. Conjecture (Mixmaster attractor conjecture) . The past attractor of the type IX dynamicalsystem coincides with the Mixmaster attractor A IX (rather than being a subset thereof ). Why is this a belief and not a fact? Theorem 6.1 implies that A − ⊆ A IX ; however, it is believedthat A − = A IX = K ∪B vac .N ∪B vac .N ∪B vac .N . (The usage of the terminology ‘Mixmaster attractor’for the set A IX reflects the strong belief in the Mixmaster conjecture.) It is difficult to imaginehow the Mixmaster attractor conjecture could possibly be violated. For instance, it seemsrather absurd that the past attractor consists only of (a subset of) heteroclinic cycles—butthere exist no proofs. A closely related belief is the following stronger statement. Conjecture.
For almost all Bianchi type IX solutions γ the α -limit set α ( γ ) coincides withthe Mixmaster attractor A IX . We use the term ‘almost all’ in a noncommittal way without specifying the measure; recall thatthe word ‘generic’ already has the well-defined meaning of ‘not past asymptotically self-similar’.(The usage of the word ‘generic’ is in accord with [26].)
Stochastic beliefs
This subsection is concerned with the (open) question of which role the Mixmaster/Kasnermap plays in the asymptotic evolution of type IX solutions. The basis for our considerationsare the results of Section 5 where we discussed the Mixmaster/Kasner map and the stochasticaspects of (generic) Kasner sequences. The
Mixmaster stochasticity conjecture supposes thatthese stochastic properties carry over to almost every type IX orbit when represented as anapproximate Kasner sequence.
Conjecture (Mixmaster stochasticity) . The approximate Kasner sequence (ˇ u l ) l ∈ N associatedwith a generic type IX orbit admits a stochastic interpretation in terms of the probability dis-tribution associated with the Kasner map, cf. Section 5. (This holds with probability one, i.e.,for almost every generic type IX orbit.) The Mixmaster stochasticity conjecture is based on a rather suggestive simple idea: Type IXevolution is like trying to follow a path of an (infinite) network of paths while the ground isshaking randomly, and where the shaking subsides with time but never stops. A type IX orbit
MIXMASTER BELIEFS l ) l ∈ N . Let (P (0) l ) l ∈ N be the exact Mixmaster sequence with P (0)0 = ˇP and consider the Taub-adapted neighborhood of this sequence associated with some prescribedvalue ǫ >
0. Finite Mixmaster shadowing entails that there exists a finite piece (ˇP l ) l =0 , ,...,L − of the approximate sequence that is contained in the prescribed Taub-adapted neighborhoodof the exact sequence (P (0) l ) l ∈ N . However, at l = L , the approximate Mixmaster sequence(ˇP l ) l ∈ N leaves the prescribed tolerance interval due to the accumulation of errors. Hence, at l = L we reset the system and consider the exact Mixmaster sequence (P (1) l ) l ≥ L with initialdata P (1) L = ˇP L . The approximate Mixmaster sequence (ˇP l ) l ≥ L is contained in the Taub-adapted neighborhood of the exact sequence (P (1) l ) l ≥ L up to l = L −
1. At l = L anotherreadjustment becomes necessary. Iterating this procedure and concatenating the finite pieces(P ( i ) l ) l = L i ,...,L i +1 − we are able to construct a sequence (¯P l ) l ∈ N with the property that the ap-proximate sequence (ˇP l ) l ∈ N is contained within the Taub-adapted neighborhood (associatedwith ǫ ) of (¯P l ) l ∈ N for all l ∈ N . The sequence (¯P l ) l ∈ N is a piecewise exact Mixmaster sequence;it is exact in intervals [ L i , L i +1 ). The length δL i = L i +1 − L i of these intervals grows beyondall bounds as i → ∞ , because shadowing takes place with an increasing degree of accuracy.The error (of the order ǫ ) between the approximate sequence (ˇP l ) l ∈ N and the piecewise ex-act sequence (¯P l ) l ∈ N results from the accumulation of numerous small errors. This obliteratesthe deterministic origin of the problem and generates a ‘randomness’ that leads to stochasticproperties. Accordingly, we expect that the exact sequences (P ( i ) l ) l ≥ L i from which (¯P l ) l ∈ N isbuilt constitute a random sample of Mixmaster sequences and thus truly reflect the stochasticproperties of the Mixmaster map. As a consequence, although the sequence (¯P l ) l ∈ N is only apiecewise exact Mixmaster sequence, it possesses the same stochastic properties as a genericMixmaster sequence. Extrapolating this line of reasoning we are able to complete the argu-ment and find that the approximate sequence (ˇP l ) l ∈ N itself reflects the stochastic propertiesof the Mixmaster map. To emphasize this aspect of stochasticity of (ˇP l ) l ∈ N we use the term‘randomized approximate sequence’. Some comments are in order.In our discussion we have assumed implicitly that the approximate sequence (ˇP l ) l ∈ N we considercan shadow any exact Mixmaster sequence for a finite number of transitions only. This is notnecessarily the case. A type IX orbit whose α -limit set is one of the heteroclinic cycles (if suchan orbit exists!) is an obvious counterexample: For every ǫ > L ∈ N such that ˇP l is contained in the Taub-adapted ǫ -neighborhood of the Mixmaster sequence associated withthe cycle. However, we expect that this type of ‘infinite shadowing’ holds at most for orbits ofa set of measure zero.A more serious limitation of the intuitive picture that we have sketched is illustrated by thefollowing related example. Consider a type IX orbit whose α -limit set is the heteroclinic cycledepicted in Figure 7(a) (where we note again that the existence of such an orbit is not proven).For the associated approximate Kasner sequence (ˇ u l ) l ∈ N we have ˇ u l → (1 + √ / l → ∞ .Consider the piecewise exact Kasner sequence (¯ u l ) l ∈ N that is associated with the piecewiseexact Mixmaster sequence. Each piece ( u ( i ) l ) l = L i ,...,L i +1 − is generated from a value u ( i ) L i =[ k ; k , k , . . . ] = [1; 1 , , , , . . . , , k ( i ) n , k ( i ) n +1 , . . . ]; we have n → ∞ as i → ∞ . It is evident that MIXMASTER BELIEFS not form a random sample of Kasner sequences. At best one could conjecture(and it is probably safe to do so) that the collection of the remainders [ k ( i ) n ; k ( i ) n +1 , . . . ] is such arandom sample. Even so, the stochastic aspect does not carry over to the approximate sequence(ˇ u l ) l ∈ N , since the approximate sequence leaves the neighborhood of each sequence ( u ( i ) l ) l = L i ,... before that sequence has entered its stochastic regime (which is characterized by the remainder[ k ( i ) n ; k ( i ) n +1 , . . . ]). Again we invoke ‘genericity’ to save the day: We conjecture that, for almost allapproximate sequences, the sequences ( u ( i ) l ) l ≥ L i are a true random sample of Kasner sequencesand that the associated stochastic properties indeed carry over to the approximate sequence(ˇ u l ) l ∈ N itself.The piecewise exact sequence (¯P l ) l ∈ N consists of pieces of length δL i = L i +1 − L i , where L i grows beyond all bounds as i → ∞ , because shadowing takes place with an increasing degreeof accuracy. However, this does not necessarily imply that δL i → ∞ as i → ∞ . The latterproperty is directly connected with the Kasner map convergence conjecture . Conjecture (Kasner map convergence) . Almost every generic type IX orbit is associated withan approximate Kasner sequence (ˇ u l ) l ∈ N such that δ ˇ u l → as l → ∞ , where δ ˇ u l describesthe error between ˇ u l +1 and the value of the Kasner parameter generated from ˇ u l by the exactKasner map (23) . The relevance of the Kasner map for the asymptotic evolution of type IX solutions rests on thevalidity of the Kasner map convergence conjecture. Let us thus give a more detailed discussionand present the line of arguments that leads to this conjecture.An orbit X ( τ ) (with segments X ( τ l ) → X ( τ l +1 ), l ∈ N ) generates a sequence of check points · · · 7→ ˇP l ˇP l +1
7→ · · · , which in turn yields a map · · · 7→ ˇ u l ˇ u l +1
7→ · · · . The parameterˇ u l +1 (associated with ˇ P l +1 ) is generated from ˇ u l (associated with ˇ P l ) by the Kasner map plusan error δ ˇ u l , see Figure 8. The magnitude of the error δ ˇ u l depends on the initial data ofthe segment, i.e., on X ( τ l ); equivalently, we may view δ ˇ u l as a function depending on (i) theposition of ˇP l on K , and (ii) the vector X ( τ l ) − ˇP l , which is orthogonal to K at ˇP l .To obtain an estimate for δ ˇ u l we introduce the order of magnitude of the error, which we denoteby ∆ˇ u l . We define ∆ˇ u l to be the average of | δ ˇ u l | over all vectors X ( τ l ) − ˇP l of equal lengththat are orthogonal to K ; alternatively, we use the somewhat ‘safer’ definition of ∆ˇ u l as themaximum of | δ ˇ u l | . By design, the order of magnitude ∆ˇ u l of the error is a function of twovariables: (i) the position of ˇP l on K , which is invariantly represented by ˇ u l , and, instead of X ( τ l ) − ˇP l itself, (ii) the (orthogonal) distance of X ( τ l ) from ˇP l (or, equivalently, from K ), i.e., k X ( τ l ) − ˇP l k (= k X ( τ l ) − K k ), which we denote by ˇ δ l , see Figure 8; in brief, ∆ˇ u l = (∆ˇ u )(ˇ u l , ˇ δ l ).In the following we investigate the behavior of the function (∆ˇ u )(ˇ u, ˇ δ ) under variations of thetwo arguments.(i) (∆ˇ u )(ˇ u, · ). Keep ˇ u fixed (and assume that ˇ u lies in the interval ˇ u ∈ (1 , ∞ ) so that its imageunder the Kasner map is finite). Then ∆ˇ u is a function of the distance ˇ δ such that ∆ˇ u → δ → u )( · , ˇ δ ). Keep the distance ˇ δ fixed, so that ∆ˇ u is a function of ˇ u . The fundamentalobservation is that ∆ˇ u becomes unbounded as (a) ˇ u → ∞ , and (b) ˇ u →
1. Case (a) is dueto the intricacies of the flow in the vicinity of the non-transversally hyperbolic Taub points(where u = ∞ ); recall that ˇ u − measures the angular distance of the check point from thenearest Taub point. For (b) we consider the orbit Q α → T α (for some α ) which correspondsto u = 1 u = ∞ . Suppose that ˇP l coincides with Q α (i.e., ˇ u l = 1). In general, ˇP l +1 will notcoincide with T α (independently of the choice of ˇ δ = ˇ δ l ); therefore ∆ˇ u l = (∆ˇ u )(ˇ u l , ˇ δ l ) = ∞ . IfˇP l is very close to Q α , which means that ˇ u l is close to 1, a small deviation of X ( τ ) from thetype II transition emanating from ˇP l will still be small at the end point X ( τ l +1 ); however, in the MIXMASTER BELIEFS α even a small deviation can translate to a large error δ ˇ u l betweenˇ u l +1 and (ˇ u l − − , which in turn results in the asserted blow-up of the order of magnitude ofthe error. The qualitative properties of ∆ˇ u as a function of ˇ δ and ˇ u are depicted in Figure 9.PSfrag replacements ˇ u = 1 ˇ u ∆ˇ u ˇ δ Figure 9: We consider a segment of an orbit X ( τ ) with initial check point ˇP associated with theKasner value ˇ u . (In accordance with the standard convention for the Mixmaster/Kasner map,the terms ‘initial’ and ‘final’ refer to a past-directed time.) The quantity ˇ δ denotes the initialdistance of the orbit from the Kasner circle, which coincides with its distance from ˇP. Thefinal check point is not generated from ˇP via the Mixmaster map, but with an error representedby δ ˇ u . The figure gives a schematic depiction of the order of magnitude ∆ˇ u of the error independence on ˇ u and ˇ δ . Each curve represents (∆ˇ u )( · , ˇ δ ), i.e., ∆ˇ u as a function of ˇ u for aconstant value of the initial distance ˇ δ ; the lower curve is associated with a smaller value of ˇ δ ,the top curve with a larger value.Consider a generic type IX orbit X ( τ ) and its representation as an approximate sequence oftransitions. By Theorem 6.1 each orbit converges to the Mixmaster attractor. As a consequencethe distance ˇ δ l = k X ( τ l ) − K k (= k X ( τ l ) − ˇP l k ) converges to zero as l → ∞ . Therefore, asˇ δ l → l → ∞ ), the sequence of functions (∆ˇ u )( · , ˇ δ l ) of Figure 9 converges to zero. However,this convergence is merely pointwise and not uniform; this behavior is crucial to understandthe behavior of δ ˇ u l as l → ∞ .If a type IX orbit X ( τ ) converges to one of the heteroclinic cycles on the Mixmaster attrac-tor, then there exists ε such that ˇ u l ∈ (1 + ε, ε − ) for all sufficiently large l . On this in-terval the function ∆ˇ u of Figure 9 converges to zero uniformly. We therefore obtain that∆ˇ u l = (∆ˇ u )(ˇ u l , ˇ δ l ) → δ ˇ u l → l → ∞ for this type IX orbit. However, the Mix-master attractor conjecture suggests that almost every type IX orbit X ( τ ) has an associatedapproximate Kasner sequence (ˇ u l ) l ∈ N that is unbounded, hence the general case is not so clear.Since (ˇ u l ) l ∈ N enters the intervals [1 , ε ) and ( ε − , ∞ ) for any ε , ∆ˇ u l = (∆ˇ u )(ˇ u l , ˇ δ l ) (and thus δ ˇ u l ) need not necessarily converge to zero. Let us elaborate.Let κ > u )(ˇ u, ˇ δ ) ≥ κ is satisfiedif and only if ˇ u ≤ f κ (ˇ δ ) or ˇ u ≥ g κ (ˇ δ ) for some functions f κ , g κ , which satisfy f κ (ˇ δ ) → g κ (ˇ δ ) → ∞ as ˇ δ →
0. Accordingly, for a given type IX orbit (and its associated approximatedKasner sequence ˇ u l ), we obtain∆ˇ u l = (∆ˇ u )(ˇ u l , ˇ δ l ) ≥ κ ⇔ ˇ u l ≤ f κ (ˇ δ l ) (35)or ˇ u l ≥ g κ (ˇ δ l ). We call the union of the two intervals (cid:0) , f κ (ˇ δ l ) (cid:1) and (cid:0) g κ (ˇ δ l ) , ∞ (cid:1) the ‘hazardzone’ (associated with epoch number l ); obviously, the hazard zone is decreasing with l . Hence, MIXMASTER BELIEFS u l , is larger than κ at epoch number l if and only if theapproximate Kasner parameter ˇ u l falls into the hazard zone (associated with l ).In connection with the Kasner map convergence conjecture the question is how often (ˇ u l ) l ∈ N enters the hazard zone, i.e., how often (35) occurs: finitely many times or infinitely many times?Suppose that the Mixmaster stochasticity conjecture is correct. Then almost every sequence(ˇ u l ) l ∈ N admits a probabilistic interpretation in terms of the probability distribution (31). Ac-cordingly, we expect the question to turn into an example of a ‘0–1 law’: There exists twoalternatives: (i) Type IX orbits (i.e., sequences (ˇ u l ) l ∈ N ) that satisfy (35) infinitely many timesare generic (i.e., of measure 1 in the state space); type IX orbits that do not are non-generic(i.e., of measure 0); (ii) Type IX orbits that satisfy (35) infinitely many times are non-generic;orbits that do not are generic. (In both cases, the two sets, the ‘generic set’ and the ‘non-generic set’, are probably non-empty.) Which of these alternatives is actually realized dependson the rate of decay of (ˇ δ l ) l ∈ N and the decay and growth of f κ and g κ , respectively, as ˇ δ → N N N , cf. (10),and N N + N N + N N , cf. [26]; this leads to the expectation that the r.h. side of (35)represents a rapidly decaying function. To conclude our line of arguments in favor of theKasner map convergence conjecture, we note that the actual error δ ˇ u l can be estimated by∆ˇ u l . Therefore, if ∆ˇ u l < κ for all l except for a finite set, then also δ ˇ u l < κ for all l exceptfor a finite set. Since κ is an arbitrary positive number, the statement of the Kasner mapconvergence conjecture ensues.We round off this section with some further remarks on the conjectures. First, we note that onemight be led to suppose that there could in fact exist type IX solutions for which the statementof the Kasner map convergence conjecture is violated, i.e., δ ˇ u l l → ∞ . In any case, theclass of these special solutions is at most of measure zero. Second, we note that the statementof the Kasner map convergence conjecture, i.e., δ ˇ u l → l → ∞ ), and the statement δL i → ∞ ( i → ∞ ), cf. the discussion on piecewise exact sequences, are expressions of one and the samestochastic property. This emphasizes that the two ‘stochastic’ conjectures tightly intertwine.Third, we remark that it is conceivable that ‘almost every’ in the two conjectures might bein fact ‘every’. In that case, the non-generic examples of Kasner/Mixmaster sequences, seeSection 5, would not have any counterparts among the type IX solutions. Of particular interestin this context would be to have an answer to question (ii) of the second subsection of Section 7.If there do not exist type IX orbits whose α -limit set is one of the heteroclinic cycles, this wouldbe a strong indication in favor of ‘every’ and against ‘almost every’. If there exist type IXorbits that converge to a cycle, ‘almost every’ is the best one can aim for in the Mixmasterstochasticity conjecture.Finally, let us briefly comment on claims of stochastic and chaotic properties of Bianchi type IXasymptotic dynamics. Chaotic aspects of the Kasner map and related maps have been studiedunder various aspects [7, 8, 19, 17, 20]. However, the relevance of these maps for type IXasymptotics rests on the two conjectures in this section. If the conjectures turn out to be wrong(e.g., if there exists a generic set of solutions that converge to heteroclinic cycles), then none ofthe results on the Kasner map carry over to full type IX dynamics. Numerical investigations,see [13] and references therein, reflect Theorem 6.1 and finite shadowing, but it is implausiblethat numerics can possibly shed light on the actual asymptotic limit of type IX solutions.Numerical errors are unavoidable and random in nature; these errors generate precisely the typeof stochasticity the simulation is looking for. Accordingly, numerical studies will necessarily Additional support for the conjecture comes from toy models that reflect the instability of the Kasner mapand its consequences. We will come back to this issue in future work.
BILLIARDS AND BILLIARD BELIEFS
In this section we briefly review the Hamiltonian billiard approach to type IX dynamics in thespirit of [6, 31] and make contact with the dynamical systems approach. The metric (2b) canbe written as g = − (det g ) ˜ N dx ⊗ dx + e − b ˆ ω ⊗ ˆ ω + e − b ˆ ω ⊗ ˆ ω + e − b ˆ ω ⊗ ˆ ω , (36)where x is an arbitrary time variable that is related to proper time t by a densitized lapsefunction ˜ N according to dt = √ det g ˜ N dx ; evidently, det g = exp[ − b + b + b )]. TheHamiltonian for the orthogonal Bianchi type IX perfect fluid models is given by H = ˜ N X α,β G αβ π α π β − R det g + 2 ρ det g = 0 , (37)where R is three-curvature and ρ = ρ (det g ) − (1+ w ) / , cf. the remark following (14) (where werecall that d log(det g ) = 6 dτ ). G αβ is the inverse of the so-called minisuperspace metric G αβ , X α,β G αβ v α w β := − X γ = δ v γ w δ = X α v α w α − (cid:16) X α v α (cid:17)(cid:16) X β w β (cid:17) , (38)i.e., G αβ is a 2 + 1-dimensional Lorentzian metric. The gravitational potential U G = − R det g is given by U G = (cid:16) e − b + e − b + e − b − e − b + b ) − e − b + b ) − e − b + b ) (cid:17) ; (39)where we have set the structure constants ˆ n α to one; the potential for the fluid is given by U F = 2 ρ det g = 2 ρ exp[ − (1 − w )( b + b + b )]. For further details see, e.g., [37] or [10,Chapter 10]; note that b α = − β α , which is used in these references; see also [30] and [31].As argued in [4, 31], b α is expected to be timelike in the asymptotic regime, i.e., P α G αβ b α b β <
0. Assuming that b α is timelike allows to introduce new metric variables instead of { b , b , b } .Defining ¯ ρ = − P α G αβ b α b β and orthogonal angular metric variables, collectively denoted by γ , leads to X α,β G αβ db α db β = − d ¯ ρ + ¯ ρ d Ω h , (40)where d Ω h is the standard metric on hyperbolic space. Making a further change of variables, λ = log ¯ ρ = log (cid:16) − X α,β G αβ b α b β (cid:17) , (41)yields X α,β G αβ π α π β = − π ρ + (¯ ρ ) − π γ = (¯ ρ ) − (cid:2) − π λ + π γ (cid:3) . (42)Choosing the lapse according to ˜ N = ¯ ρ leads to a Hamiltonian of the form H = (cid:2) − π λ + π γ (cid:3) + ¯ ρ X A c A exp( − ρ w A ( γ )) , (43) BILLIARDS AND BILLIARD BELIEFS w A ( γ ) denotes certain linear forms of the variables γ α , i.e., w A ( γ ) = P β w A β γ β ; see [31]for details.The essential point is that one expects that ¯ ρ → ∞ towards the singularity and hence thateach term ¯ ρ exp[ − ρw A ( γ )] becomes an infinitely high sharp wall described by an infinitestep function Θ ∞ ( x ) that vanishes for x < x ≥
0. Accordingly, only‘dominant’ terms in the potential are assumed to be of importance for the generic asymptoticdynamics, while ‘subdominant’ terms, i.e., terms whose exponential functions can be obtainedby multiplying dominant wall terms, are neglected. In the present case there are three dominantterms in U G (which is the minimal set of terms required to define the billiard table), the threeexponentials exp( − b α ). Dropping the subdominant terms in the limit ¯ ρ → ∞ leads to anasymptotic Hamiltonian of the form H ∞ = (cid:2) − π λ + π γ (cid:3) + X A =1 Θ ∞ ( − w A ( γ )) , (44)where only the three dominant terms appear in the sum. The correspondence between thedynamical systems picture and the Hamiltonian picture is easily obtained by noting that theHamiltonian constraint (37) is proportional to the Gauss constraint (9). The dominant termscorrespond to the terms N α , α = 1 , ,
3, in (9); the subdominant terms are collected in ∆ II .The non-trivial dynamics described by (44) resides in the variables γ , i.e., in the hyperbolicspace. It can be described asymptotically as geodesic motion in hyperbolic space constrained bythe existence of sharp reflective walls, i.e., the asymptotic dynamics is determined by the type IX‘billiard’ given in Figure 10. Based on the heuristic considerations the limiting Hamiltonian (44)is believed to describe generic asymptotic dynamics.PSfrag replacements γ γ γ Figure 10: The Bianchi type IX billiard: The disc represents hyperbolic space. The asymptoticdescription of a solution is given by geodesic billiard motion inside the triangle, which acts asa stationary infinite potential wall, yielding a ‘configuration space picture’ of the asymptoticdynamics.The Hamiltonian picture (as represented by the billiard) and the dynamical systems picture (asrepresented by the Mixmaster attractor) are ‘dual’ to each other. In the Hamiltonian picturethe asymptotic dynamics is described as the evolution of a point in the billiard. Straightlines (geodesics) in hyperbolic space correspond to Kasner states. Wall bounces correspond toBianchi type II solutions; the bounces change the Kasner states according to the Kasner map.
CONCLUDING REMARKS . The walls in the Hamiltonian billiardare translated to motion in the dynamical systems picture—Bianchi type II heteroclinic orbits;these type II transitions yield exactly the same rule for changing Kasner states as the wallbounces: the Kasner map. Since the variables Σ α are proportional to π α , it is natural torefer to the projected dynamical systems picture as a ‘momentum space’ representation of theasymptotic dynamics.Summing up, ‘walls’ and ‘straight line motion’ switch places between the Hamiltonian formu-lation and dynamical systems description of asymptotic dynamics, and the two pictures giveequivalent complementary asymptotic pictures.The above heuristic derivation of the limiting Hamiltonian rests on two basic assumptions: Itis assumed that b α is timelike in the asymptotic regime and that one can drop the subdom-inant terms. These assumptions correspond to assuming that Ω and ∆ II can be set to zeroasymptotically, i.e., the procedure precisely assumes Theorem 6.1. (The Hamiltonian analysisin [46, Chapter 2] uses the same assumptions, and hence the present discussion is of directrelevance for that work as well.) That these assumptions are highly non-trivial is indicated bythe difficulties that the proof of Theorem 6.1 has presented; cf. [26, 27]. An alarming exampleis Bianchi type VIII; in this case the heuristic procedure in this respect leads to exactly thesame asymptotic results, but so far there exist no proof that Ω and ∆ II tend to zero towardsthe singularity for generic solutions. We elaborate on the differences between type VIII andtype IX in [27]. Moreover, like in the state space picture there exists no proof that all of thepossible billiard trajectories are of relevance for the asymptotic dynamics of type IX solutions.For example, there exists a correspondence between periodic orbits in the dynamical systemspicture and the Hamiltonian billiard picture, and it is not excluded a priori that solutions areforced to one, or several, of these. We emphasize again that all proposed measures of chaosthat take billiards as the starting point, see [46] and references therein, rely the conjecturedconnection between the Mixmaster map and asymptotic type IX dynamics.The above discussion shows that there are non-trivial assumptions and subtle phenomena thatare being glossed over in heuristic billiard ‘derivations.’ Nevertheless, we do believe that thebilliard procedure elegantly uncovers the main generic asymptotic features, and it might befruitful to attempt to combine the dynamical systems and Hamiltonian picture in order toprove the Mixmaster conjectures. A tantalizing hint is that in the billiard picture π λ becomesan asymptotic constant of the motion. Rewriting this dimensional constant of the motion interms of the dynamical systems variables yields π λ = 2 H p det g h − ( τ − τ ) + log (cid:0) N Σ N Σ N Σ (cid:1)i , (45)where log √ det g = 3( τ − τ ) and τ is a constant. The purpose of this paper is two-fold. On the one hand, we analyze the main known results onthe asymptotic dynamics of Bianchi type IX vacuum and orthogonal perfect fluid models to-wards the initial singularity. The setting for our discussion is the Hubble-normalized dynamicalsystems approach, since this is essentially the set-up which has led to the first rigorous state-ments on Bianchi type IX asymptotics [26]. (We choose slightly different variables to emphasize
EFERENCES − / or VIII; this suggests that the situation in the general inhomogeneous case is even morecomplicated than expected. Furthermore, numerical studies are incapable of shedding light onthe asymptotic limit. This is mainly due to the accumulation of inevitable random numericalerrors that make it a priori impossible to track a particular type IX orbit. Finally, although theHamiltonian methods are a formidable heuristic tool, so far this approach has not yielded anyproofs about asymptotics. Nevertheless, it might prove to be beneficial to explore the possiblesynergies between dynamical systems and Hamiltonian methods.In this paper and in [27, 30] we have encountered remarkable subtleties as regards the asymp-totic dynamics of oscillatory singularities; this emphasizes the importance of a clear distinctionbetween facts and beliefs. Acknowledgments
We thank Alan Rendall, Hans Ringstr¨om, and in particular Lars Andersson for useful discus-sions. We gratefully acknowledge the hospitality of the Mittag-Leffler Institute, where part ofthis work was completed. CU is supported by the Swedish Research Council.
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