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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 essentially understood ; ( ii ) bianchi type  ix is believed to be a role model that captures the generic features of generic spacelike singularities . however , we will illustrate in this paper that there are reasons to question these beliefs . the idea that type  ix is essentially understood is a misconception . in actuality , surprisingly little is known , i.e. , proved , about type  ix asymptotic dynamics ; at the same time there exist widely held , but rather vague , beliefs about mixmaster dynamics , oscillations , and chaos , which are frequently mistaken to be facts . there is thus a need for clarification : what are the known facts and what is merely believed about type  ix asymptotics ? we will address this issue in two ways : 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 the limitations of these results . on the other hand , we will provide the infrastructure that makes it possible to sharpen commonly held beliefs ; based on this framework we will formulate explicit refutable conjectures .    historically , bianchi type ix vacuum and orthogonal perfect fluid models entered the scene in the late sixties through the work of belinskii , khalatnikov and lifshitz @xcite and misner and chitr  @xcite . bkl attempted to understand the detailed nature of singularities and were led to the type ix models via a rather convoluted route , while misner was interested in mechanisms that could explain why the universe today is almost isotropic . bkl and misner independently , by means of quite different methods , reached the conclusion that the temporal behavior of the type ix models towards the initial singularity can be described by sequences of anisotropic kasner states , i.e. , bianchi type i vacuum solutions . these sequences are determined by a discrete map that leads to an oscillatory anisotropic behavior , which motivated misner to refer to the type ix models as mixmaster models  @xcite . this discrete map , the kasner map , was later shown to be associated with stochasticity and chaos  @xcite , a property that has generated considerable interest  and confusion , see , e.g. ,  @xcite and references therein . a sobering thought : all claims about chaos in einstein s equations rest on the ( plausible ) belief that the kasner map actually describes the asymptotic dynamics of einstein 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 manner that allowed one to apply powerful dynamical systems techniques  @xcite ; gradually a picture of a hierarchy of invariant subsets emerged where monotone functions restricted the asymptotic dynamics to boundaries of boundaries , see  @xcite and references therein . based on work reviewed and developed in  @xcite and by rendall  @xcite , ringstrm eventually produced the first major proofs about asymptotic type  ix dynamics  @xcite . this achievement is remarkable , but it does not follow 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 the role of type  ix models in the context of generic singularities been established  @xcite . the outline of the paper is as follows . in section [ basic ] we briefly describe the hubble - normalized dynamical systems approach and establish the connection with the metric approach . for simplicity 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 a linear equation of state . in section  [ subsets ] we discuss the levels of the bianchi type  ix so - called lie contraction hierarchy of subsets , where we focus on the bianchi type  i and type  ii subsets . in section  [ nongeneric ] we present the results of the local analysis of the fixed points of the dynamical system and discuss the stable and unstable subsets of these points which are associated with non - generic asymptotically self - similar behavior . section  [ maps ] is devoted to a study of the network of sequences of heteroclinic orbits ( heteroclinic chains ) that is induced by the dynamical system on the closure of the bianchi type  ii vacuum boundary of the type  ix state space ( which we refer to as the mixmaster attractor subset ) . these sequences of orbits are associated with the mixmaster map , which in turn induces the kasner map and thus the kasner sequences . we analyze the properties of non - generic kasner sequences and discuss the stochastic properties of generic sequences . in section  [ furthermix ] we discuss the main ` mixmaster facts ' : ringstrm s ` bianchi type  ix attractor theorem '  @xcite , theorem  [ rinthm ] , and a number of consequences that follow from theorem  [ rinthm ] and from the results on the mixmaster / kasner map . in addition , we introduce and discuss the concept of ` finite mixmaster shadowing ' . in the subsection ` attractor beliefs ' of section  [ stochasticbeliefs ] we formulate two conjectures that reflect commonly held beliefs about type  ix asymptotic dynamics and list some open issues that are directly connected with these conjectures . in the subsection ` stochastic beliefs ' we address the open question of which role the mixmaster / kasner map and its stochastic properties actually play in type  ix asymptotic dynamics . this culminates in the formulation , and discussion , of two ` stochastic ' conjectures . in section  [ billiard ] we present the hamiltonian billiard formulation , see  @xcite or  @xcite ; we demonstrate that this approach yields a ` dual ' formulation of the asymptotic dynamics . we point out that the billiard approach is a formidable heuristic picture , but fails to turn beliefs into facts . we conclude in section  [ concl ] with a discussion of the main themes of this paper . throughout this paper we use units so that @xmath0 and @xmath1 , where @xmath2 is the speed of light and @xmath3 the gravitational constant . 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 of state ; we require the energy conditions ( weak / strong / dominant ) to hold , i.e. , @xmath4 and @xmath5 where @xmath6 , and where @xmath7 and @xmath8 are the energy density and pressure of the fluid , respectively . by we exclude the special cases @xmath9 and @xmath10 , where the energy conditions are only marginally satisfied . , see  @xcite . the case @xmath10 is known as the stiff fluid case , for which the speed of sound is equal to the speed of light . the asymptotic dynamics of stiff fluid solutions is simpler than the oscillatory behavior characterizing the models with range @xmath11 , and well understood  @xcite . ( in the terminology introduced below , the stiff fluid models are asymptotically self - similar . ) we will therefore refrain from discussing the stiff fluid case in this paper . ] as is well known , see , e.g. ,  @xcite and references therein , for these models there exists a symmetry - adapted frame @xmath12 ,    [ bixform ] @xmath13    hence , the type  ix models naturally belong to the so - called class  a bianchi models , see table  [ classamodels ] . let @xmath14 where @xmath15 . furthermore , define @xmath16 where @xmath17 denotes the second fundamental form associated with   of the sh hypersurfaces @xmath18 . the quantities @xmath19 and @xmath20 can be interpreted as the expansion and the shear , respectively , of the normal congruence of the sh hypersurfaces . in a cosmological context it is customary to replace @xmath19 by the hubble variable @xmath21 ; this variable is related to changes of the spatial volume density according to @xmath22 . evidently , in bianchi type  ix ( and type viii ) there is a one - to - one correspondence between the ` orthonormal frame variables ' @xmath23 ( with @xmath24 ) and @xmath25 ; in particular , the metric @xmath26 is obtained from @xmath27 via  . ( for the lower bianchi types  i@xmath28 , some of the variables @xmath27 are zero , cf . ; in this case , the other frame variables , i.e. , @xmath29 , are needed as well to reconstruct the metric ; see  @xcite for a group theoretical approach . ) .the class  a bianchi types are characterized by different signs of the structure constants @xmath30 , where @xmath31 is any permutation of @xmath32 . in addition to the above representations there exist equivalent representations associated with an overall change of sign of the structure constants ; e.g. , another type ix representation is @xmath33 . [ cols="^,^,^,^",options="header " , ]     for all class  a models except type ix the constraint   implies that @xmath34 ; in bianchi type  ix , however , @xmath35 is possible . for type ix , define @xmath36 employing   and   and using that @xmath37 + \delta \geq 0\:,\ ] ] where equality holds iff @xmath38 , we find @xmath39 the function @xmath40 is strictly monotonically increasing along orbits of bianchi type  ix . to see this we use   and compute @xmath41 \delta\:,\ ] ] where we note that @xmath42 because of the constraints . in combination with   it follows that @xmath43 and @xmath44 are _ bounded towards the past_.    the right hand side of the reduced system  ( [ ixeq ] ) consists of polynomials of the state space variables and is thus a regular dynamical system . solutions of   of bianchi types  i  viii are global in @xmath45 , since   implies the bounds @xmath34 and @xmath46 which control the evolution of @xmath47 in  . solutions of   of bianchi type  ix are global towards the past , since @xmath48 is bounded from below ; this follows from  , which yields @xmath49 , so that @xmath50 . the decoupled equation for @xmath51 , cf .  , yields that @xmath52 as @xmath53 , because @xmath54 is non - negative . since @xmath54 is bounded as @xmath55 , the asymptotics of @xmath51 can be bounded by exponential functions from above and below . it follows that the equation @xmath56 can be integrated to yield @xmath57 as a function of @xmath45 such that @xmath58 as @xmath55 . in addition to   it is useful to also consider an auxiliary equation for the matter quantity @xmath44 , @xmath59\omega\:.\ ] ] making use of   and   we conclude that for all orthogonal perfect fluid models with a linear equation of state we have @xmath60\tau\right)$ ] , and hence @xmath61 as @xmath53 , which yields a past singularity . the divergence of @xmath7 can also be directly read off from the matter equation @xmath62 . the bianchi type  ix state space @xmath63 is characterized by the conditions @xmath64 , @xmath65 , @xmath66 . we write @xmath67 . the notation is such that the subscript denotes the non - zero variables among @xmath68 . setting one or more of these variables to zero ( which corresponds to lie contractions  @xcite ) yields invariant boundary subsets which describe more special bianchi types . since the type ix models exhibit discrete symmetries associated with axes permutations , the contractions generate all possible representations of the more special ( lie contracted ) bianchi types ( which are associated with such permutations ) : the bianchi type  @xmath28 subspace @xmath69 is given by the disjoint union of three equivalent sets , @xmath70 , where , e.g. , @xmath71 denotes the type  @xmath28 subset with @xmath72 , @xmath73 and @xmath74 ; the bianchi type  @xmath75 subspace @xmath76 by the union @xmath77 ; the bianchi type  @xmath78 subspace @xmath79 by @xmath80 . note that the bianchi type  @xmath81 subspace does not appear as a boundary subset of @xmath63 . a bianchi subset contraction diagram for type ix is given in figure  [ contraction ] . [ cc][cc]@xmath82 [ cc][cc]@xmath71 [ cc][cc]@xmath83 [ cc][cc]@xmath84 [ cc][cc]@xmath85 [ cc][cc]@xmath86 [ cc][cc]@xmath87 [ cc][cc]@xmath88 [ rc][cc]@xmath63 [ rc][cc]@xmath69 [ rc][cc]@xmath76 [ rc][cc]@xmath79    each set of the lie contraction hierarchy is the union of an invariant vacuum subset , i.e. , @xmath89 , and an invariant fluid subset , i.e. , @xmath90 . to refer to a vacuum [ fluid ] subset of a bianchi set @xmath91 we use the notation @xmath92 [ @xmath93 . in this spirit , e.g. , the type  ii subset decomposes as @xmath94 .    in the following we analyze the boundary subsets to the extent needed in order to understand the asymptotic type  ix dynamics . the bianchi type  i subset is given by @xmath95 , @xmath96 , @xmath74 and @xmath97 ; since @xmath98 , @xmath99 , @xmath100 vanish , we denote this subset by @xmath88 , cf . figure  [ contraction ] . the _ vacuum subset _ consists of a circle of fixed points_the kasner circle _ @xmath101 , which is characterized by @xmath102 . it is common to represent different points on @xmath101 in terms of the kasner exponents @xmath103 , @xmath104 each fixed point on @xmath101 represents a kasner solution ( kasner metric ) with the corresponding exponents . the kasner circle is divided into six equivalent sectors , denoted by permutations of the triple @xmath32 , where sector @xmath31 is characterized by @xmath105 , see figure  [ kasnercirc ] . the boundaries of the sectors are six special points that are associated with solutions that are locally rotationally symmetric ( lrs ) : @xmath106 are given by @xmath107 or @xmath108 and yield the three equivalent lrs solutions whose intrinsic geometry is non - flat ; the _ taub points _ @xmath109 are given by @xmath110 or @xmath111 and correspond to the flat lrs solutions  the taub representation of minkowski spacetime . [ cc][cc]@xmath112 [ cc][cc]@xmath113 [ cc][cc]@xmath114 [ cc][cc]@xmath32 [ cc][cc]@xmath115 [ cc][cc]@xmath116 [ cc][cc]@xmath117 [ cc][cc]@xmath118 [ cc][cc]@xmath119 [ cc][cc]@xmath120 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][cc]@xmath124 [ cc][cc]@xmath125 [ cc][cc]@xmath126 [ cc][cc]@xmath127 [ cc][cc]@xmath128 [ cc][cc]@xmath129    the quantity @xmath130 ( or , equivalently , @xmath131 ) is invariant under changes of the axes and thus naturally captures the ` physical essence ' of a solution independent of the chosen frame ; however , it is typically replaced by the _ kasner parameter _ @xmath132 through @xmath133\:.\ ] ] the kasner parameter @xmath132 parameterizes the kasner exponents uniquely ( up to the permutation symmetry ) ; we have @xmath134 for sector @xmath31 of @xmath101 , where @xmath135 . therefore , each point on sector @xmath31 is represented by a unique value of @xmath136 . at the boundary points of sector @xmath31 , which are @xmath106 and @xmath137 , the kasner parameter is @xmath138 and @xmath139 , respectively . permuting @xmath31 yields a physically equivalent state on a different sector ; accordingly , each @xmath135 represents an equivalence class of six points on @xmath140 . in contrast , @xmath138 describes the three points @xmath141 ; @xmath139 yields @xmath142 .    while the bianchi type  i vacuum subset coincides with the kasner circle @xmath140 , which is given by @xmath143 , the bianchi type  i perfect fluid subset is the set @xmath144 . from   it is straightforward to deduce that there exists a central fixed point , the _ friedmann fixed point _ @xmath145 , given by @xmath146 @xmath147 , which corresponds to the isotropic friedmann - robertson - walker ( frw ) solution . solutions with @xmath148 are given by radial straight lines originating from @xmath140 and ending at @xmath145 . these results rely on the assumption @xmath149 , see  . let us consider the bianchi type  ii subset @xmath150 given by @xmath151 , @xmath152 . in this case , the @xmath153-direction is singled out , while there exists a discrete symmetry associated with the interchange of the @xmath154- and @xmath155-direction . the lrs subset @xmath156 is a subset of codimension one which divides the state space into two equivalent parts , the subsets @xmath157 and @xmath158 ( related by permuting the @xmath154- and @xmath155-axes ) .    since the past singularity is of particular interest in our considerations , it is convenient to choose the time direction towards the past , i.e. , to use a reversed time variable @xmath159 according to @xmath160 which we do in the remainder of this section ; accordingly , approach to the past singularity means @xmath161 . ( in section  [ maps ] we will see that vacuum type  ii orbits are the building blocks for the mixmaster map and the closely related kasner map ( bkl map ) . it is a well established convention that forward iterations of these maps are directed towards the singularity . to agree with this convention the use of a past - directed time variable in a discussion of bianchi type  ii models thus suggests itself . )    on @xmath150 , the gauss constraint @xmath162 can be used to replace @xmath163 by @xmath44 as a dependent variable . the system   thus becomes @xmath164\,,\ ] ] where @xmath165 and @xmath166 , @xmath167 ; we have @xmath168 . let us first consider the vacuum subset @xmath169 , i.e. , @xmath89 . there do not exist any fixed points in the type  ii vacuum subset @xmath169 , but the boundary of the vacuum subset coincides with the kasner circle @xmath140 . the orbits of   form a family of straight lines in @xmath169 , where each orbit connects one fixed point on @xmath140 with another fixed point on @xmath140 ; hence each orbit is heteroclinic  @xcite . following the nomenclature of  @xcite we call these heteroclinic orbits bianchi type  ii _ transitions _ , because each orbit can be viewed as representing a transition from one kasner state to another . we denote these transitions by @xmath170 , where each @xmath170 emanates from @xmath171 . if the initial point is a point of sector @xmath172 , then the final point is a point of @xmath173 ; interchanging @xmath154 and @xmath155 yields the transitions emanating from @xmath174 ; if the initial point is @xmath175 , the final point is @xmath137 ; the points @xmath109 and @xmath176 are not connected with any other fixed point ( they are ` fixed points ' under the present ` type ii map ' ) , see figure  [ typeii ] . [ cc][cc]@xmath112 [ cc][cc]@xmath113 [ cc][cc]@xmath114 [ cc][cc]@xmath120 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][cc]@xmath124 [ cc][cc]@xmath125 [ cc][cc]@xmath126    let @xmath177 denote the initial fixed point on @xmath140 ; this point can be represented in terms of the kasner parameter @xmath178 by using  . the orbit ( transition ) emanating from this fixed point is given in terms of an auxiliary function @xmath179 by    [ zeqs ] @xmath180 + \eta\ , \sigma^{\mathrm{i}}_{\alpha/\beta } \ : , \qquad \sigma_{\gamma } = -4 [ 1 - \eta ] + \eta \,\sigma^{\mathrm{i}}_{\gamma}\:,\ ] ] where @xmath181 is determined by the equation @xmath182    and the conditions that @xmath183 and @xmath184 . the quantity @xmath185 is in the interval @xmath186 for @xmath187 ; @xmath188 corresponds to @xmath189 and describes the orbit @xmath190 ; @xmath139 corresponds to @xmath191 and describes the ` isolated ' points @xmath192 , @xmath193 . since @xmath181 increases from @xmath194 to @xmath185 as @xmath159 goes from @xmath195 to @xmath196 , @xmath185 is called the growth factor  @xcite . in section [ maps ] , the transitions @xmath170 , as represented by  , will appear as the building blocks for the mixmaster / kasner map .    while there do not exist any fixed points in the vacuum subset of @xmath150 , there exists one fixed point in @xmath150 with @xmath197 , the _ collins - stewart fixed point _ @xmath198 , which corresponds to one representation of the lrs solutions found by collins and stewart  @xcite . @xmath198 is given by @xmath199 and @xmath200 ( which yields @xmath201 ) . the fixed point @xmath198 is the source ( w.r.t . @xmath159 ) for all orbits in @xmath150 with @xmath90 . in the limit @xmath202 all solutions in @xmath203 converge to fixed points on the bianchi type  i boundary of @xmath150 : there exists one orbit ( which corresponds to an lrs solution ) that converges to @xmath145 as @xmath204 ; every other orbit converges to a fixed point on @xmath205 or @xmath206 on @xmath140 , or to @xmath137 ( in the lrs case ) . for a detailed discussion of these results see  @xcite .      in anticipation of theorem  [ rinthm ] which implies that generic orbits of bianchi type  ix do not have @xmath154-limit , the @xmath154-limit set @xmath208 of a point @xmath209 is defined as the set of all accumulation points towards the past ( i.e. , as @xmath53 ) of the orbit @xmath210 through @xmath211 . the simplest examples of @xmath154-limit sets are fixed points and periodic orbits . ] points ( w.r.t.the standard future directed time variable @xmath45 ) on any of the bianchi type  @xmath28 subsets @xmath71 , @xmath83 , @xmath212 , we refrain from giving a detailed discussion of these subspaces here . ( however , note that in order to _ prove _ theorem  [ rinthm ] , a detailed understanding of solutions of bianchi type  @xmath28 is essential ; in fact , in the proof of theorem  [ rinthm ] bianchi type  @xmath28 is ubiquitous ; we refer to  @xcite and  @xcite . ) in the present context it suffices to note that on each subset @xmath213 there exists a line of fixed points @xmath214 given by @xmath215 and @xmath216 ( so that @xmath89 ) . since @xmath214 emanates from the point @xmath217 we call it the ` taub line . ' like @xmath137 itself , each of fixed points on @xmath214 is associated with a representation of minkowski spacetime ( in an lrs type @xmath28 symmetry foliation ) . in the previous section we have given the fixed points associated with the system   on @xmath218 . a local dynamical systems analysis of the fixed points shows whether or not these points attract type  ix orbits in the limit @xmath53 .. ] we merely state the results here and refer to  @xcite for details .    * each fixed point @xmath219 on @xmath220 is a transversally hyperbolic saddle ( one stable instead of @xmath45 , we must replace ` stable ' by ` unstable ' . ] mode , see figure  [ unstable ] , and three unstable modes ; the stable manifold of @xmath219 coincides with a vacuum type  ii transition orbit , see figure  [ typeii ] ) . the taub points @xmath221 are center saddles with a two - dimensional unstable manifold and a three - dimensional center manifold , where @xmath109 is excluded as an @xmath154-limit set on the center manifold , see  @xcite ; consequently there do not exist any type  ix solutions that converge to any of the points on @xmath140 as @xmath53 . * the fixed point @xmath145 on @xmath222 is a hyperbolic saddle ( with @xmath222 as a two - dimensional stable manifold and an additional three - dimensional unstable manifold ) . accordingly , @xmath145 attracts a two - parametric family of type  ix orbits as @xmath53 . these solutions have a so - called isotropic singularity . * the fixed points @xmath223 ( @xmath224 ) on @xmath225 are hyperbolic saddles ( with @xmath225 as a three - dimensional stable manifold and an additional two - dimensional unstable manifold ) . the unstable modes are associated with the equations @xmath226 ( for @xmath227 ) . therefore , each of the fixed points @xmath223 attracts an ( equivalent ) one - parameter set of type  ix orbits in the limit @xmath53 . * each fixed point on @xmath228 on @xmath229 is a center saddle . on the three - dimensional center manifold ( which coincides with @xmath230 , @xmath231 ) the point acts as a ( non - hyperbolic ) sink , see  @xcite . since there is a two - dimensional unstable manifold , there exists , for each fixed point on @xmath228 , a one - parameter family of type  ix orbits that converges to it as @xmath53 ; these orbits correspond to lrs solutions . ( conversely , generic lrs type  ix solutions converge to @xmath228 , see , e.g.,@xcite . )    [ cc][cc]@xmath112 [ cc][cc]@xmath113 [ cc][cc]@xmath114 [ cc][cc]@xmath98 [ cc][cc]@xmath99 [ cc][cc]@xmath99 [ cc][cc]@xmath100 [ cc][cc]@xmath100 [ cc][cc]@xmath98 [ cc][cc]@xmath120 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][cc]@xmath124 [ cc][cc]@xmath125 [ cc][cc]@xmath126 [ cc][cc]@xmath127 [ cc][cc]@xmath128 [ cc][cc]@xmath129    collecting the results we see that the solutions whose @xmath154-limit is one of the fixed points form a subfamily of measure zero of the ( four - parameter ) family of bianchi type  ix solutions . following the nomenclature of @xcite we thus refer to these solutions as _ non - generic _ solutions of bianchi type  ix . alternatively , to capture the asymptotic behavior of these solution , we use the term _ past asymptotically self - similar _ solutions . ( since a fixed point in the hubble - normalized dynamical systems formulation corresponds to a self - similar solution , see e.g.  @xcite , solutions that converge to a fixed point are asymptotically self - similar . ) the past asymptotically self - similar solutions comprise the lrs bianchi type  ix solutions . as seen above , generic lrs solutions converge to @xmath228 towards the past ( and each solution that converges to @xmath228 is lrs ) , but there exist exceptional lrs solutions that converge to @xmath145 or @xmath223 . the remaining orbits whose limit point is either @xmath145 or @xmath223 correspond to past asymptotically self - similar solutions that are non - lrs . clearly , every solution that converges to @xmath145 or @xmath223 is a non - vacuum solution , since @xmath232 at @xmath145 and @xmath223 . it is natural to ask how the non - generic orbits are embedded in the state space @xmath218 . the lrs orbits form the three lrs subsets @xmath233 , which are the hyperplanes given by the conditions @xmath234 , @xmath235 , where @xmath236 . the orbits whose @xmath154-limit set is the fixed point @xmath223 ( for some @xmath154 ) form the set @xmath237 in @xmath63 ; we call @xmath237 the collins - stewart manifold . the local analysis of the fixed point @xmath223 and the regularity of the dynamical system   imply that the collins - stewart manifold @xmath237 is a two - dimensional surface ; it can be viewed as a two - dimensional manifold with boundary embedded in @xmath218 ( where this boundary corresponds to an orbit in @xmath238 ) . analogously , the orbits whose @xmath154-limit set is the fixed point @xmath145 form the set @xmath239 in @xmath63 , which we call the isotropic singularity manifold , since solutions converging to @xmath145 are those with an isotropic singularity . the isotropic singularity manifold @xmath239 is a three - dimensional hypersurface ; 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 constitute examples for asymptotic self - similarity breaking ; for other such examples , see  @xcite . the central theme in this paper is the past asymptotic behavior of the generic models . the _ mixmaster attractor _ @xmath240 ( alternatively referred to as the bianchi type  ix attractor ) is defined to be the subset of @xmath218 given by the union of the bianchi type  i and  ii vacuum subsets , i.e. , @xmath241 . since the type  ii vacuum subset consists of three equivalent representations we obtain @xmath242 in this section we investigate the structures that the flow of dynamical system   induces on @xmath240 . in particular we discuss the mixmaster map , the kasner map , and the era map . to agree with the well - established convention for these maps , the direction of time will be taken towards the past . in section  [ subsets ] we have seen that the vacuum type  ii orbits , i.e. , the orbits on @xmath243 , @xmath224 , are heteroclinic orbits that emanate from and converge to fixed points on the kasner circle @xmath140 , see figures  [ typeii ] and  [ alltypeii ] . in accord with section  [ subsets ] we refer to these orbits as _ transitions _ and denote them by @xmath244 , @xmath245 . [ rc][cc][0.9][0]@xmath120 [ tl][tl][0.9][0]@xmath121 [ bl][bl][0.9][0]@xmath122 [ lc][cc][0.9][0]@xmath123 [ br][br][0.9][0]@xmath124 [ tr][tr][0.9][0]@xmath125   @xmath246 , @xmath247 on the type  ii subsets @xmath248 , @xmath249 , @xmath250 onto @xmath251-space . concatenation of these transition orbits yields sequences of transitions . the arrows indicate the direction of time towards the past.,title="fig:",scaledwidth=25.0% ] @xmath246 , @xmath247 on the type  ii subsets @xmath248 , @xmath249 , @xmath250 onto @xmath251-space . concatenation of these transition orbits yields sequences of transitions . the arrows indicate the direction of time towards the past.,title="fig:",scaledwidth=25.0% ] @xmath246 , @xmath247 on the type  ii subsets @xmath248 , @xmath249 , @xmath250 onto @xmath251-space . concatenation of these transition orbits yields sequences of transitions . the arrows indicate the direction of time towards the past.,title="fig:",scaledwidth=25.0% ]    the type  ii transitions are the building blocks for the analysis of the mixmaster attractor . by concatenating transitions we obtain a _ sequence of transitions _ , also known as a _ heteroclinic chain_. since each fixed point on @xmath140 ( except for the taub points ) is the initial value for one single transition , see figures  [ unstable ] and  [ alltypeii ] , the concatenation of transitions is unique : each fixed point ( except for the taub points ) generates a unique sequence of transitions . note , however , that the ` direction of time ' is relevant . for each fixed point @xmath252 on @xmath140 , which is not one of the taub points , there exists one single transition emanating from @xmath252 at @xmath253 , but there are two transitions converging to @xmath252 as @xmath202 . therefore , concatenating transitions in the reversed direction of time leads to ambiguities . ( in terms of the standard future - directed time variable @xmath45 we have the converse statement : it is possible to make unambiguous retrodictions , but not predictions . ) let @xmath254 and let @xmath255 denote the initial point of the @xmath256 transition ( @xmath257 is also the end point of the @xmath258 transition ) . we refer to the sequence @xmath259 of kasner fixed points , which is induced by the sequence of transitions , as being generated by the  _ mixmaster map_. the mixmaster map can be visualized by a map in @xmath251-space , obtained by inscribing @xmath101 in a triangle with corners at @xmath260 and cyclic permutations , from which the ( projections of the ) transition orbits ` originate ' as straight lines ; see figure  [ mixmaster ] . [ cc][cc]@xmath112 [ cc][cc]@xmath113 [ cc][cc]@xmath114 [ cc][cc]@xmath120 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][cc]@xmath124 [ cc][cc]@xmath125 [ cc][cc]@xmath126 [ cc][cc]@xmath127 [ cc][cc]@xmath128 [ cc][cc]@xmath129    let the initial kasner state of a transition be represented by the kasner parameter @xmath261 , where we assume @xmath262 , since neither of the taub points @xmath120 , @xmath121 , @xmath122 can be the initial value for a transition . inserting   and   into we find that a transition maps the kasner parameter @xmath263 to the parameter @xmath264 , where @xmath265 ( u^{\mathrm{i } } - 1)^{-1 } & \qquad   \text{if } \quad u^{\mathrm{i } } \in [ 1,2]\ : . \end{array}\right.\ ] ] the information contained in this _ kasner map _ suffices to represent the collection of all transition orbits ( as a whole ) . however , for each particular mapping @xmath266 , there exist six ( equivalent ) associated transitions ; there exist only three ( equivalent ) associated transitions : @xmath267 , @xmath268 , @xmath269 . ] this is simply because @xmath263 characterizes the initial kasner fixed point on @xmath140 only up to permutations of the axes . hence , in order to reconstruct a particular transition from  , we supplement   with information about the initial sector of the transition , which determines the position of the axes .    in terms of the kasner parameter , a sequence of transitions corresponds to an iteration of  . let @xmath254 and let @xmath270 denote the initial kasner state of the @xmath256 transition . this transition maps @xmath270 to @xmath271 , i.e. , @xmath272 ( u_l - 1)^{-1 } & \qquad   \text{if } \quad u_l \in [ 1,2]\ , . \end{array}\right.\ ] ] we refer to this map as the ( iterated ) _ kasner map _ ( which is also known as the bkl map  @xcite ) . since each value of the kasner parameter @xmath273 represents an equivalence class of six kasner fixed points , the kasner map can be regarded as the map induced by the mixmaster map on these equivalence classes via the equivalence relation .    in a sequence @xmath274 that is generated by the kasner map  , each kasner state @xmath270 is called an _ epoch_. every sequence @xmath274 possess a natural partition into pieces ( which contain a finite number of epochs each ) where the kasner parameter is monotonically decreasing according to the simple rule @xmath275 ; these pieces are called _ eras _  @xcite . an era begins with a maximal value @xmath276 of the kasner parameter ( where @xmath276 is generated from @xmath277 by @xmath278^{-1}$ ] ) , continues with a sequence of kasner parameters obtained via @xmath279 , and ends with a minimal value @xmath280 that satisfies @xmath281 , so that @xmath282^{-1}$ ] begins a new era . @xmath283 let us denote the initial ( @xmath284 maximal ) value of the kasner parameter @xmath132 in era number @xmath285 ( where @xmath286 ) by @xmath287 . following  @xcite we decompose @xmath287 into its integer part @xmath288 $ ] and its fractional part @xmath289 , i.e. , @xmath290\ : , \quad    x_s= \ { { \mathsf{u}}_s\}\:.\ ] ] the number @xmath291 represents the ( discrete ) length of era @xmath285 , which is simply the number of kasner epochs it contains . the final ( @xmath284 minimal ) value of the kasner parameter in era @xmath285 is given by @xmath292 , which implies that era number @xmath293 begins with @xmath294 the map @xmath295 is ( a variant of ) the so - called _ era map _ ; starting from @xmath296 it recursively determines @xmath287 , @xmath297 , and thereby the complete kasner sequence @xmath298 . the era map admits a straightforward interpretation in terms of continued fractions . consider the continued fraction representation of the initial value , i.e. , @xmath299\:.\ ] ] the fractional part of @xmath300 is @xmath301 $ ] ; since @xmath302 is the reciprocal of @xmath303 we find @xmath304\:.\ ] ] therefore , the era map is simply a shift to the left in the continued fraction expansion , @xmath305 \:\mapsto\ :    { \mathsf{u}}_{s+1 } = [ k_{s+1 } ; k_{s+2 } , k_{s+3},\dotsc]\:.\ ] ]    the properties of the kasner sequence depend on the initial value @xmath296 . if and only if the initial kasner parameter @xmath306 is a rational number , i.e. , if and only if @xmath307 , then its continued fraction representation is finite , i.e. ,    [ cfki ] @xmath308\ : ,      \tag{\ref{cfki}i}\ ] ] where @xmath309 . therefore , there exists only a finite number of eras ( where the last one begins with @xmath310 ) , and the kasner sequence is finite . at the end of era number @xmath311 , the kasner parameter reaches @xmath138 , which subsequently terminates the recursion  . since @xmath312 is a set of measure zero in @xmath313 , this case is non - generic . a quadratic irrational ( quadratic surd ) is an algebraic number of degree @xmath314 , i.e. , an irrational solution of a quadratic equation with integer coefficients . if and only if the initial kasner parameter @xmath306 is a quadratic irrational , i.e. , if and only if @xmath315 , where @xmath316 and @xmath317 is not a perfect square ( i.e. , @xmath318 ) , then its continued fraction representation is periodic , i.e. , @xmath319\ : ;      \tag{\ref{cfki}ii}\ ] ] the notation is such that the part in parenthesis , i.e. , @xmath320 , is repeated ad infinitum . is a quadratic irrational such that @xmath321 , then @xmath322 $ ] , i.e. , the continued fraction is purely periodic without any preperiod . ] consequently , the era map becomes periodic ( after the @xmath311 era ) , and we thus obtain a periodic sequence of eras and a periodic kasner sequence @xmath274 . it is straightforward to see that while the period of the era sequence is @xmath323 , the period of the kasner sequence is @xmath324 ; see the examples below . since the set of algebraic numbers of degree two ( or equivalently the set of equations with integer coefficients  it is a subset of @xmath325 ) is a countable set , case ( ii ) is also non - generic . an irrational number is called badly approximable if its markov constant , let @xmath326 denote the distance from @xmath211 to the nearest integer , i.e. , @xmath327 . the markov constant @xmath328 of a number @xmath329 is defined as @xmath330 , see  @xcite . it is known that @xmath331 for all @xmath332 . ] is finite . if and only if @xmath306 is badly approximable , then the coefficients ( partial quotients ) in its continued fraction representation are bounded , i.e. , @xmath333 \qquad \text{with}\quad k_i \leq k\:\ , \forall i     \tag{\ref{cfki}iii}\ ] ] for some positive constant @xmath334 . consequently , the sequence of eras and the kasner sequence @xmath274 are bounded , i.e. , @xmath335 @xmath336 . obviously , case ( ii ) is a subcase of case ( iii ) . ( note , however , that there is probably no relationship between case ( iii ) and algebraic numbers of degree greater than  @xmath314 , since it is expected that these numbers are well approximable . ) the set of badly approximable numbers are a set of lebesgue measure zero , hence this case is non - generic .    if and only if the initial kasner parameter @xmath306 is a well approximable irrational number , then the partial quotients @xmath337 in the continued fraction representation @xmath338      \tag{\ref{cfki}iv}\ ] ]    are unbounded ( and we can construct a diverging subsequence from the sequence of partial quotients @xmath339 ) . this is the generic case , and hence generically the kasner sequence @xmath274 is infinite and unbounded .    in terms of continued fractions , the kasner sequence @xmath340 generated by @xmath341 $ ] is @xmath342 \rightarrow   \big [ k_0 -1 ; k_1 , k_2 , \dotsc \big ] \rightarrow \big [ k_0 -2 ; k_1 , k_2 ,   \dotsc \big ] \rightarrow \ldots \rightarrow \big[1 ; k_1 , k_2 , \dotsc \big ] \\ \rightarrow \mathsf{u}_1 & = \big [ k_1 ; k_2 , k_3 , \dotsc \big ]   \rightarrow   \big [   k_1 - 1 ; k_2 , k_3 , \dotsc \big ] \rightarrow \big [ k_1 -2 ; k_2 , k_3 , \dotsc \big ] \rightarrow \ldots \rightarrow \big[1 ; k_2 , k_3 , \dotsc \big ] \\ \rightarrow \mathsf{u}_2 & = \big [ k_2 ; k_3 , k_4 , \dotsc \big ]   \rightarrow   \big [   k_2 - 1 ; k_3 , k_4 , \dotsc \big ] \rightarrow \big [ k_2 -2 ; k_3 , k_4 , \dotsc \big ] \rightarrow \ldots\end{aligned}\ ] ] let us give some examples for periodic _ era sequences _ and _ kasner sequences_. if @xmath343 = ( 1+\sqrt{5})/2 $ ] , which is the golden ratio , then @xmath344 @xmath345 . it follows that the kasner sequence is also a sequence with period @xmath194 , @xmath346 if @xmath347 = 1 + \sqrt{2}$ ] , then @xmath348 @xmath345 ; hence the era sequence is a sequence of period @xmath194 . however , the associated kasner sequence has period @xmath314 , @xmath349 analogously , the initial value @xmath350 = ( 3+\sqrt{13})/2 $ ] generates an era sequence of period @xmath194 and an associated kasner sequence of period @xmath351 . finally let @xmath352 = 1 + \sqrt{3/2 } \simeq 2.2247 $ ] . then the era sequence has period @xmath314 with @xmath353 \simeq 2.2247 $ ] for even @xmath311 and @xmath354 \simeq 4.4495 $ ] for odd @xmath311 . the associated kasner sequence @xmath355 has period @xmath356 , @xmath357 in the state space description of sequences ( in terms of the mixmaster map ) , an _ epoch _ is simply a point @xmath257 on the kasner circle . ( it is one of the six points in the equivalence class associated with the kasner parameter @xmath270 . ) transitions connect epochs and thus generate the mixmaster map . the kasner parameter @xmath132 can be employed to measure the ( angular ) distance of a point @xmath252 on @xmath140 from the taub points or from the non - flat lrs points : if @xmath358 , then @xmath252 is at an angular distance of approximately @xmath359 from one of the taub points . on the other hand , in the vicinity of the non - flat lrs points ( where @xmath360 ) , @xmath361 is a linear measure for the angular distance of @xmath252 from the closest of the non - flat lrs points . if @xmath362 , then @xmath252 is closer to one of the non - flat lrs points @xmath106 than to any of the taub points @xmath109 . therefore , in the state space picture , an _ era _ can be described as a ( finite ) sequence of points @xmath363 , @xmath364 , obtained from the mixmaster map , whose angular distance from the taub points is monotonically increasing . the era begins with a fixed point @xmath365 that is close to one of the taub points @xmath109 ( the preceding point @xmath366 was closer to one of the other two taub points @xmath176 , @xmath367 , than to @xmath109 ) ; then the distance from the taub point @xmath109 monotonically increases until , for @xmath368 , it exceeds the distance to the non - flat lrs points ; this is the terminal point for the era ; it is connected by the following transition with the initial point of the next era ; a good illustration is figure  [ period3 ] , 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 sequences in the state space picture ; this is exemplified by the heteroclinic cycles in figure  [ period1a ] and  [ period1b ] . the state space description of the cases ( i)(iv ) , which are characterized by the initial kasner parameter @xmath296 , is the following :    * iff @xmath307 , see  , the mixmaster sequence of kasner fixed points @xmath369 is finite . after a finite number of transitions , at the end of era @xmath311 , the sequence reaches one of the lrs points @xmath106 ( where @xmath138 ) ; a last transition follows , namely the transition @xmath370 , and the sequence terminates in one of the taub points . * iff @xmath306 is a quadratic surd , i.e. , @xmath315 for some @xmath371 where @xmath372 is not a perfect square , see  , then the mixmaster sequence of kasner points @xmath259 is eventually periodic , where the period is a multiple of @xmath324 ; see figure  [ cycle ] . viewed as a periodic sequence of transitions ( which are heteroclinic orbits ) we obtain a heteroclinic cycle . in figure [ cycle ] we give some of the heteroclinic cycles associated with kasner sequences with periods @xmath194 , @xmath314 , and @xmath351 in their projection onto @xmath373-space . note that due to permutation symmetry there are several cycles associated with a given periodic kasner sequence . * iff @xmath306 is a badly approximable irrational number , see  , there exists a neighborhood of the taub points @xmath109 such that the mixmaster sequence of kasner points @xmath259 does not enter this neighborhood . this is simply because there exists a maximal value of the sequence @xmath355 . * iff @xmath306 is a well approximable irrational number , see  , then the mixmaster sequence @xmath259 comes arbitrarily close to the taub points . this is the generic case . [ cc][cc]@xmath112 [ cc][cc]@xmath113 [ cc][cc]@xmath114 [ cc][cc]@xmath120 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][cc]@xmath124 [ cc][cc]@xmath125 +    in the following we analyze the generic case ( iv ) in more detail . let @xmath306 be a well approximable irrational number , i.e. , a number whose continued fraction expansion @xmath374\ ] ] defines an unbounded sequence @xmath339 . by construction , era number @xmath375 contains @xmath337 epochs ( which we call its length ) . a natural question to ask concerns the distribution of the partial quotients @xmath337 . for a ` typical ' well approximable irrational number , how often does the number @xmath194 appear in the sequence @xmath339 ? how often the number @xmath314 ? and what about the number @xmath376 ? the answer is given by khinchin s law  @xcite . let @xmath377 denote the probability that a randomly chosen partial quotient among @xmath378 equals @xmath379 . in the asymptotic limit , i.e. , for @xmath380 we have @xmath381 i.e. , the partial quotients of the continued fraction representation of   are distributed like a random variable whose probability distribution is given by  . ( by  , the number @xmath194 appears in @xmath382 of the slots , the number @xmath314 in @xmath383 , and the number @xmath376 in @xmath384 of the slots . ) khinchin s law applies for _ almost all _ numbers @xmath306 .    for a ( generic ) kasner sequence @xmath355 with initial parameter @xmath385 $ ] and its associated era sequence @xmath386 , where @xmath387 $ ] , the expression @xmath388 of   represents the probability that a randomly chosen era of @xmath386 has length @xmath389 ; this corresponds to the probability that the initial value of an era is contained in the interval @xmath390 . in this manner , the probability distribution   makes possible a stochastic interpretation of generic kasner sequences . the probability distribution   results in extraordinary properties of the mixmaster / kasner map , which will be of crucial importance in the considerations of section  [ beliefs ] . we do not discuss details here but refer to future work ; however , we can not refrain from giving a teaser : for a generic kasner sequence @xmath355 and its associated era sequence @xmath386 there exist infinitely many eras such that the length ( i.e. , the number of epochs ) of the @xmath311 era is larger than @xmath391 ; however , for sufficiently large @xmath311 , the length is guaranteed to be bounded by @xmath392 . ( for a proof of this result by borel and bernstein see  @xcite ; see also  @xcite . ) properties of this kind underline the remarkable intricacies of the heteroclinic structures on the mixmaster attractor . in this section we turn to what is known about the past asymptotic dynamics of generic type  ix solutions . the main mixmaster fact is ringstrm s ` bianchi type  ix attractor theorem ' . consider a solution of bianchi type  ix that is either vacuum or associated with a perfect fluid satisfying @xmath393 . recall that such a solution is called _ generic _ if it is not past asymptotically self - similar , i.e. , if its @xmath154-limit set is neither the point @xmath145 , nor any of the points @xmath223 , nor a point on @xmath228 ; in other words , a generic solution corresponds to an orbit in @xmath63 that is neither contained in @xmath239 , nor in @xmath237 , nor in @xmath233 . therefore , the set of generic bianchi type  ix states is an open set in @xmath63 . ( to conform with  @xcite we use the future directed time variable @xmath45 . ) the main results concerning generic bianchi type  ix models are due to ringstrm  @xcite ; these results rest on earlier work that is reviewed and derived in  @xcite , and on  @xcite . in the following we state the main theorem in a version adapted to our purposes . [ rinthm ] let @xmath394 be a generic solution of bianchi type  ix , i.e. , a generic solution of   in @xmath63 . then @xmath395 as @xmath396 . note that this theorem applies to both the fluid and the vacuum case ; in the latter case   becomes @xmath397 and @xmath398 . the proof of theorem  [ rinthm ] given in  @xcite is delicate . in the first part it is proved that the @xmath154-limit set of a generic bianchi type  ix solution is non - empty and must contain a point on the kasner circle . the main part of the proof deals with the fact that the function @xmath399 is in general not monotone . there exist times where @xmath399 increases ( as @xmath53 ) ; the associated growth must therefore be controlled and shown to be negligible compared to the overall decrease in @xmath400 . this is done by a careful analysis of the equations and ( approximate ) solutions . in  @xcite we give an alternative and relatively short and succinct proof of theorem  [ rinthm ] which is based on an in - depth understanding of the hierarchical structure of the dynamical system   ( as represented by figure  [ contraction ] ) . it is important to note , however , that either of the proofs fail in the other bianchi types that are conjectured to exhibit an oscillatory approach towards the singularity , i.e. , the proofs fail for types  @xmath401 and viii . this is unfortunate , since there are reasons to believe that these models are more relevant than type  ix as regards the dynamics of generic ( inhomogeneous ) cosmologies , see  @xcite . using the concept of the mixmaster attractor , cf .  , we obtain an equivalent formulation of theorem  [ rinthm ] : let @xmath402 be a generic solution of bianchi type  ix . then @xmath403 where the distance @xmath404 is given as @xmath405 . theorem  [ rinthm ] thus states that the attractor of generic type  ix solutions resides on @xmath240 ; however , whether the past attractor is in fact @xmath240 or merely a subset thereof remains open . ( the terminology ` mixmaster attractor ' is seductive but might turn out to be quite misleading . ) likewise , the theorem does not provide any direct information about the details of the asymptotic behavior of solutions . the prerequisite for a deeper understanding of the asymptotic behavior and the oscillatory dynamics of generic type  ix solutions is an understanding of the mixmaster attractor . in section  [ maps ] we have identified the structures on the mixmaster attractor @xmath240 that are induced by the flow of the dynamical system : heteroclinic cycles [ case ( ii ) ] and finite [ case ( i ) ] and infinite heteroclinic sequences [ cases ( iii ) and ( iv ) ] . all these structures qualify ( a priori ) as possible @xmath154-limit sets of generic type  ix orbits . in conjunction with these results , the main theorem  [ rinthm ] implies a number of further facts about the attractor`mixmaster attractor facts ' , which we give as a list of * corollaries*. ( for proofs see  @xcite , and also  @xcite . )    1 . [ possessesalpha ] a generic , i.e. not past asymptotically self - similar , type  ix orbit possesses an @xmath154-limit point on @xmath240 . [ onethenall ] if @xmath406 is an @xmath154-limit point of a type  ix orbit , then the entire heteroclinic cycle / sequence ( mixmaster sequence ) through @xmath252 must be contained in the @xmath154-limit set . [ taubthenmany ] if one of the taub points @xmath407 is an @xmath154-limit point of a type  ix orbit , then the @xmath154-limit set contains kasner fixed points associated with arbitrarily large values of the kasner parameter @xmath132 . 4 .   for generic solutions of bianchi type  ix the weyl curvature scalar @xmath408 ( and therefore also the kretschmann scalar ) becomes unbounded towards the past . 5 .   taking into account both the expanding and contracting phases of bianchi type  ix solutions , generic bianchi type  ix initial data generate an inextendible maximally globally hyperbolic development associated with past and future singularities where the curvature becomes unbounded . [ uniformcor ] convergence to the mixmaster attractor is uniform on compact sets of generic initial data : let @xmath409 be a compact set in @xmath63 that does not intersect any of the manifolds @xmath239 , @xmath237 , @xmath233 , so that each initial data @xmath410 generates a generic type  ix solution . let @xmath411 denote the type  ix solution with @xmath412 . then @xmath413 uniformly in @xmath410 . corollary 3 implies that the @xmath154-limit set contains an infinite set of kasner fixed points in a neighborhood of the taub point(s ) , but this set is not necessarily a continuum of fixed points ; cf .  the previous discussion about possible @xmath154-limit sets . if a kasner point with @xmath414 is contained in the @xmath154-limit of an orbit , so is a taub point . this is an immediate consequences of the results of section  [ maps ] , case ( i ) . on the other hand , if the @xmath154-limit set of an orbit is a heteroclinic cycle or a heteroclinic sequence ( possibly in combination with a cycle ) associated with cases ( ii ) and ( iii ) of section  [ maps ] , then there exists a neighborhood of the taub points whose intersection with the @xmath154-limit set is empty . note that the oscillatory behavior of asymptotic type  ix dynamics , which we unfortunately know no details about , constitutes an example of asymptotic self - similarity breaking  @xcite . in order to make progress as regards the details of the asymptotic oscillatory behavior , it is natural to first establish the connection between the mixmaster / kasner map and dynamics for a finite time interval @xmath415 , discussed next .      to make contact between the mixmaster map and bianchi type  ix asymptotic dynamics we introduce 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 initial data of the sequence so that the type  ix solution generated by this data remains close to the sequence for some ` time ' let @xmath416 be a kasner fixed point ( but @xmath417 ) and let @xmath306 be the associated value of the kasner parameter . there exists a unique sequence of transitions @xmath418 and an associated mixmaster sequence @xmath419 with @xmath416 as initial data ; the associated kasner sequence is @xmath355 . ( if @xmath307 , the sequence terminates at one of the taub points after a finite number of transitions . ) we shall make the definition that a type  ix solution shadows a finite piece @xmath420 of the sequence of transitions if it is contained in a prescribed ( small ) tubular neighborhood of @xmath421 . however , a standard @xmath422-neighborhood of the sequence fails to be a reasonable measure of closeness , because in the vicinity of a taub point the transitions lie so dense that consecutive transitions are not separated from each other by their respective @xmath422-neighborhoods . therefore , the introduction of _ adapted _ tubular neighborhoods is necessary to take into account the sensitivity of the flow at the taub points and to capture more accurately the intuitive idea of shadowing . let @xmath423 be small . a ` _ _ taub - adapted neighborhood _ _ ' of the mixmaster sequence @xmath424 is a sequence @xmath425 of open balls , where @xmath426 is centered at the point @xmath257 and has radius @xmath427 , i.e. , @xmath428 . ( the radius @xmath429 is chosen to ensure that the intersection of the ball @xmath426 with the kasner circle induces more or less a standard @xmath422-neighborhood @xmath430 of the kasner parameter @xmath270 ; recall from section  [ maps ] that @xmath359 measures the angular distance of a fixed point from a taub point . ) a taub - adapted tubular neighborhood of the sequence of transitions @xmath418 is the sequence of tubes @xmath431 that linearly interpolate between @xmath426 and @xmath432 . based on this definition we say that a type  ix solution @xmath433 _ shadows _ a finite piece @xmath420 of the sequence of transitions if it moves in a prescribed taub - adapted tubular neighborhood @xmath421 , i.e. , if there exists a sequence of times @xmath434 such that @xmath433 is contained in @xmath435 for all @xmath436 $ ] for all @xmath437 .    making use of these concepts , a formulation of finite mixmaster shadowing is the following : let @xmath423 and @xmath438 . consider the sequence of transitions @xmath439 emanating from an initial kasner point @xmath416 and its taub - adapted tubular neighborhood ( associated with @xmath422 ) . then there exists @xmath440 such that each type  ix orbit @xmath433 that is generated by initial data @xmath441 with @xmath442 shadows the finite piece @xmath421 of the sequence of transitions . ( a proof of this statement  in a slightly different form  has been given by rendall  @xcite . alternatively , one can invoke the regularity of the dynamical system , the center manifold reduction theorem and continuous dependence on initial data . ) evidently , @xmath443 depends on the choice of @xmath422 and @xmath444 . more importantly , however , @xmath443 depends on the position of @xmath416shadowing is not uniform ; in particular , if we consider a series of initial points that approach one of the taub points , then @xmath443 necessarily converges to zero along this series  shadowing is more delicate in the vicinity of a taub point . we will return to this issue in some detail in the next section . finite mixmaster shadowing concerns any generic type  ix orbit @xmath433 . let @xmath445 be an @xmath154-limit point of the type  ix orbit @xmath433 ; without loss of generality we may assume that @xmath252 is not one of the taub points . ( the existence of such a point is ensured by corollaries  [ possessesalpha][taubthenmany ] of section  [ furthermix ] . ) for simplicity we assume that @xmath252 is associated with an irrational value of the kasner parameter , which guarantees that the sequence @xmath259 emanating from @xmath446 is an infinite sequence . since @xmath252 is an @xmath154-limit point of @xmath433 , there exists a sequence of times @xmath447 , @xmath448 ( @xmath449 ) , such that @xmath450 ( @xmath449 ) . therefore , we observe a recurrence of phases , where the orbit @xmath433 shadows @xmath259 with an increasing degree of accuracy , i.e. , shadowing takes place for increasingly longer pieces of the sequence or in ever smaller neighborhoods . ( if the kasner parameter of @xmath252 is rational , then the sequence @xmath259 is finite and terminates at a taub point . @xmath433 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  [ beliefs ] . the concept of shadowing leads directly to the concept of _ approximate sequences _ which we introduce next . consider a generic type  ix orbit @xmath402 and the function @xmath451 , where the distance @xmath452 is given as @xmath453 . when the orbit @xmath433 traverses a ( sufficiently small ) neighborhood of a fixed point on @xmath454 , the function @xmath455 exhibits a unique local minimum . this is immediate from the transversal hyperbolic saddle structure of the fixed point . ( however , the flow in the vicinity of the taub points is more intricate , since these points are not transversally hyperbolic . ) it follows that the function @xmath451 can be used to partition @xmath433 into a sequence of segments in a straightforward manner : into segments seems natural ; it is important to note , however , that any definition depends on the formulation of the problem and is to a certain extent arbitrary . for instance , instead of using the minima of @xmath451 one might prefer to analyze the projection of the orbit onto @xmath373-space and use the extrema of @xmath456 . however , the conclusions drawn from any construction of segments are quite insensitive to the details of the definition . ] the local minima of @xmath451 form an infinite sequence @xmath457 such that @xmath458 as @xmath459 . ( this follows directly from corollaries  [ possessesalpha ] and  [ onethenall ] because @xmath433 has @xmath154-limit point(s ) on the kasner circle and @xmath154-limit points on the type  ii subset . ) a _ segment _ of @xmath433 is defined to be the solution curve between two consecutive minima , i.e. , the image of the interval @xmath460 $ ] . ( note that @xmath461 in the approach to the singularity , while the discrete ` time ' @xmath256 is past - directed , i.e. , @xmath462 towards the singularity . this convention is chosen to agree with the standard convention for the mixmaster and the kasner map . ) in the asymptotic regime , i.e. , in the approach to the mixmaster attractor , finite shadowing entails that a finite sequence of segments will resemble a finite sequence of type  ii transitions ; this assumes , 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 orbit in its segmented form an _ approximate sequence of transitions_.    in addition , we define a sequence of ` check points ' that is associated with an approximate sequence of transitions . each minimum @xmath463 of @xmath451 is associated with a kasner fixed point @xmath464 ( a ` check point ' ) that is defined as the minimizer on @xmath140 of the distance between @xmath465 and @xmath140 . ( note that the check points @xmath466 do not lie on the type  ix orbit @xmath433 . ) since each check point @xmath464 is associated with a value @xmath467 of the kasner parameter , the sequence @xmath466 induces a sequence @xmath468 . the value @xmath469 is in general not generated from @xmath467 by the exact kasner map  , but differs from that value by an error of @xmath470 . in our terminology , the sequence @xmath466 is an approximate mixmaster sequence ; @xmath468 is an approximate kasner sequence , see figure  [ checkmap ] . [ rc][rc][0.9][0]@xmath465 [ lc][lc][0.9][0]@xmath471 [ lc][lc][0.9][0]@xmath472 [ rc][rc][0.9][0]@xmath473 [ cc][cc][0.7][0]@xmath474 [ cc][cc][1][0]@xmath433 [ cc][cc][1][0]@xmath140 [ cc][cc][0.5][2]mixmaster map [ tc][tc][0.5][2]kasner map [ rc][rc][0.9][0]@xmath475 . the points @xmath465 , @xmath476 , are the local minima of the distance between @xmath433 and @xmath140 . the ` check point ' @xmath472 is the point on @xmath140 that is closest to @xmath433 ( for @xmath45 in a neighborhood of @xmath463 ) . since each check point @xmath472 is associated with a value @xmath467 of the kasner parameter , the ` approximate mixmaster sequence ' @xmath477 induces an ` approximate kasner sequence ' @xmath468 . in general , @xmath473/@xmath469 are not generated from @xmath472/@xmath467 by the exact mixmaster / kasner map  , but differ by an error @xmath478/@xmath470.,title="fig:",scaledwidth=35.0% ]    the approximate kasner sequence @xmath467 associated with a type  ix orbit @xmath433 does not follow the kasner map   exactly . a natural question to ask , however , is whether the errors @xmath470 of the approximate kasner sequence converge to zero as @xmath462 or not . if @xmath479 as @xmath459 for a type  ix orbit , this means that its dynamics is completely described by an ` asymptotic mixmaster / kasner map ' , i.e. , by a map that converges to the mixmaster / kasner map towards the singularity . however , if @xmath480 as @xmath462 , then the evolution is interrupted repeatedly  infinitely many times  by phases where the dynamics is completely different from the mixmaster dynamics , e.g. , ` eras of small oscillations ' @xcite . ( in the present state space description an era of small oscillations is associated with type  @xmath28 behavior in the vicinity of one of the taub lines @xmath228 , where @xmath481 and @xmath163 are small and of the same order ; for details we refer to the discussion of type  @xmath28 dynamics in  @xcite . ) in the subsection ` stochastic mixmaster beliefs ' of section  [ beliefs ] we will investigate the behavior of @xmath470 along type  ix orbits in detail . there remain several important open problems . in the following we will address the most pressing questions ; in particular we will transform vague beliefs into refutable conjectures , and give arguments in their favor . an immediate question is the following : what are the actual @xmath154-limit sets of type  ix orbits ? consider a type  ix orbit @xmath153 . the @xmath154-limit set @xmath482 contains a number of kasner fixed points ( and the associated type  ii transitions ) , each of which is characterized by a particular value of the kasner parameter @xmath132 . the set of kasner parameters obtained in this way from @xmath482 we denote by @xmath483 . the question of which form @xmath483 can take for different orbits @xmath153 is open , the most significant issues being the following :    * is it possible that there exists @xmath153 whose limit set @xmath483 consists of rational numbers only ? it is straightforward to exclude that @xmath483 coincides with @xmath312 itself ( or a dense subset thereof ) , the reason being that @xmath154-limit sets are necessarily closed . another a priori constraint is that @xmath483 must contain @xmath484 ( which characterizes the taub points ) if @xmath483 contains a rational number ; see section  [ maps ] . corollary  [ taubthenmany ] of section  [ furthermix ] then implies that @xmath483 is unbounded , i.e. , @xmath483 contains arbitrarily large values of the kasner parameter . a set that is compatible with these basic requirements is , e.g. , @xmath485 . whether there exist orbits @xmath153 such that @xmath483 takes this ( or a related ) form remains open . * is it conceivable that there exist orbits @xmath153 such that @xmath482 is a heteroclinic cycle , see figure  [ cycle ] ? in this case the set @xmath483 is generated by a quadratic surd @xmath486 $ ] via the kasner map ; see section  [ maps ] ; in particular , @xmath483 is finite . however , whether orbits @xmath153 with this particular past asymptotic behavior really exist is an open problem . * can there exist orbits @xmath153 such that @xmath483 is bounded but contains infinitely many @xmath132-values ? is the kasner sequence generated by a badly approximable number a candidate ? the kasner sequence @xmath355 generated by a badly approximable number is an infinite sequence that is bounded . however , there must be at least one accumulation point of this sequence ; if @xmath483 contains @xmath355 , then @xmath483 must contain the accumulation point of the sequence as well ( since @xmath154-limit sets are closed ) . if this accumulation point is a well approximable number , @xmath483 can not be bounded ; however , if the accumulation point is a quadratic surd , no inconsistencies arise . hence , a priori , there might exist orbits @xmath153 such that @xmath483 is not finite but still bounded . whether this is indeed the case is doubtful , but hard to exclude a priori . an open problem that might be quite separate from the questions raised above concerns the behavior of all type  ix orbits save a set of measure zero . the past attractor of a dynamical system given on a state space @xmath487 is defined as the smallest closed invariant set @xmath488 such that @xmath489 for all @xmath490 apart from a set of measure zero @xcite .    the past attractor of the type  ix dynamical system coincides with the mixmaster attractor @xmath240 ( rather than being a subset thereof ) .    why is this a belief and not a fact ? theorem  [ rinthm ] implies that @xmath491 ; however , it is believed that @xmath492 . ( the usage of the terminology ` mixmaster attractor ' for the set @xmath240 reflects the strong belief in the mixmaster conjecture . ) it is difficult to imagine how the mixmaster attractor conjecture could possibly be violated . for instance , it seems rather absurd that the past attractor consists only of ( a subset of ) heteroclinic cycles  but there exist no proofs . a closely related belief is the following stronger statement .    for almost all bianchi type  ix solutions @xmath153 the @xmath154-limit set @xmath482 coincides with the mixmaster attractor @xmath240 . we use the term ` almost all ' in a noncommittal way without specifying the measure ; recall that the word ` generic ' already has the well - defined meaning of ` not past asymptotically self - similar ' . ( the usage of the word ` generic ' is in accord with  @xcite . ) this subsection is concerned with the ( open ) question of which role the mixmaster / kasner map plays in the asymptotic evolution of type  ix solutions . the basis for our considerations are the results of section  [ maps ] where we discussed the mixmaster / kasner map and the stochastic aspects of ( generic ) kasner sequences . the _ mixmaster stochasticity conjecture _ supposes that these stochastic properties carry over to almost every type  ix orbit when represented as an approximate kasner sequence . the approximate kasner sequence @xmath468 associated with a generic type  ix orbit admits a stochastic interpretation in terms of the probability distribution associated with the kasner map , cf .  section  [ maps ] . ( 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  ix evolution is like trying to follow a path of an ( infinite ) network of paths while the ground is shaking randomly , and where the shaking subsides with time but never stops . a type  ix orbit tries to follow a sequence of transitions on the mixmaster attractor while random errors cause the orbit to lose track . due to the errors , the orbit is incapable of following one particular sequence forever ; after a finite time , the type  ix orbit has deviated too much and it leaves the vicinity of the sequence . although , temporarily , the orbit is contained in a neighborhood of a different sequence , it is bound to lose track of that sequence as well eventually . accordingly the type  ix orbit is thrown around in the space of mixmaster sequence with the effect that the type  ix orbit inherits the stochastic properties of generic mixmaster sequences .    in the following we paint a heuristic picture that makes this idea a little more concrete . consider a type  ix orbit ( approximate sequence of transitions ) and the associated approximate mixmaster sequence @xmath477 . let @xmath493 be the exact mixmaster sequence with @xmath494 and consider the taub - adapted neighborhood of this sequence associated with some prescribed value @xmath423 . finite mixmaster shadowing entails that there exists a finite piece @xmath495 of the approximate sequence that is contained in the prescribed taub - adapted neighborhood of the exact sequence @xmath493 . however , at @xmath496 , the approximate mixmaster sequence @xmath477 leaves the prescribed tolerance interval due to the accumulation of errors . hence , at @xmath496 we reset the system and consider the exact mixmaster sequence @xmath497 with initial data @xmath498 . the approximate mixmaster sequence @xmath499 is contained in the taub - adapted neighborhood of the exact sequence @xmath497 up to @xmath500 . at @xmath501 another readjustment becomes necessary . iterating this procedure and concatenating the finite pieces @xmath502 we are able to construct a sequence @xmath503 with the property that the approximate sequence @xmath477 is contained within the taub - adapted neighborhood ( associated with @xmath422 ) of @xmath503 for all @xmath504 . the sequence @xmath503 is a piecewise exact mixmaster sequence ; it is exact in intervals @xmath505 . the length @xmath506 of these intervals grows beyond all bounds as @xmath507 , because shadowing takes place with an increasing degree of accuracy . the error ( of the order @xmath422 ) between the approximate sequence @xmath477 and the piecewise exact sequence @xmath508 results from the accumulation of numerous small errors . this obliterates the deterministic origin of the problem and generates a ` randomness ' that leads to stochastic properties . accordingly , we expect that the exact sequences @xmath509 from which @xmath503 is built constitute a random sample of mixmaster sequences and thus truly reflect the stochastic properties of the mixmaster map . as a consequence , although the sequence @xmath508 is only a piecewise exact mixmaster sequence , it possesses the same stochastic properties as a generic mixmaster sequence . extrapolating this line of reasoning we are able to complete the argument and find that the approximate sequence @xmath466 itself reflects the stochastic properties of the mixmaster map . to emphasize this aspect of stochasticity of @xmath466 we use the term ` randomized approximate sequence ' . some comments are in order .    in our discussion we have assumed implicitly that the approximate sequence @xmath466 we consider can shadow any exact mixmaster sequence for a finite number of transitions only . this is not necessarily the case . a type  ix orbit whose @xmath154-limit set is one of the heteroclinic cycles ( if such an orbit exists ! ) is an obvious counterexample : for every @xmath423 there exists @xmath510 such that @xmath464 is contained in the taub - adapted @xmath422-neighborhood of the mixmaster sequence associated with the cycle . however , we expect that this type of ` infinite shadowing ' holds at most for orbits of a set of measure zero . a more serious limitation of the intuitive picture that we have sketched is illustrated by the following related example . consider a type  ix orbit whose @xmath154-limit set is the heteroclinic cycle depicted in figure  [ period1a ] ( where we note again that the existence of such an orbit is not proven ) . for the associated approximate kasner sequence @xmath468 we have @xmath511 as @xmath459 . consider the piecewise exact kasner sequence @xmath512 that is associated with the piecewise exact mixmaster sequence . each piece @xmath513 is generated from a value @xmath514 = [ 1;1,1,1,1,\ldots,1,k^{(i)}_n , k^{(i)}_{n+1},\ldots]$ ] ; we have @xmath449 as @xmath507 . it is evident that these pieces do _ 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 @xmath515 $ ] is such a random sample . even so , the stochastic aspect does not carry over to the approximate sequence @xmath468 , since the approximate sequence leaves the neighborhood of each sequence @xmath516 before that sequence has entered its stochastic regime ( which is characterized by the remainder @xmath515 $ ] ) . again we invoke ` genericity ' to save the day : we conjecture that , for almost all approximate sequences , the sequences @xmath517 are a true random sample of kasner sequences and that the associated stochastic properties indeed carry over to the approximate sequence @xmath468 itself . the piecewise exact sequence @xmath503 consists of pieces of length @xmath506 , where @xmath518 grows beyond all bounds as @xmath507 , because shadowing takes place with an increasing degree of accuracy . however , this does not necessarily imply that @xmath519 as @xmath507 . the latter property is directly connected with the _ kasner map convergence conjecture_.    almost every generic type  ix orbit is associated with an approximate kasner sequence @xmath468 such that @xmath520 as @xmath459 , where @xmath470 describes the error between @xmath469 and the value of the kasner parameter generated from @xmath467 by the exact kasner map  . the relevance of the kasner map for the asymptotic evolution of type  ix solutions rests on the validity of the kasner map convergence conjecture . let us thus give a more detailed discussion and present the line of arguments that leads to this conjecture . an orbit @xmath433 ( with segments @xmath521 , @xmath522 ) generates a sequence of check points @xmath523 , which in turn yields a map @xmath524 . the parameter @xmath469 ( associated with @xmath525 ) is generated from @xmath467 ( associated with @xmath526 ) by the kasner map plus an error @xmath470 , see figure  [ checkmap ] . the magnitude of the error @xmath470 depends on the initial data of the segment , i.e. , on @xmath465 ; equivalently , we may view @xmath470 as a function depending on ( i ) the position of @xmath472 on @xmath140 , and ( ii ) the vector @xmath527 , which is orthogonal to @xmath140 at @xmath472 .    to obtain an estimate for @xmath470 we introduce the _ order of magnitude _ of the error , which we denote by @xmath528 . we define @xmath528 to be the average of @xmath529 over all vectors @xmath527 of equal length that are orthogonal to @xmath140 ; alternatively , we use the somewhat ` safer ' definition of @xmath528 as the maximum of @xmath529 . by design , the order of magnitude @xmath528 of the error is a function of two variables : ( i ) the position of @xmath472 on @xmath140 , which is invariantly represented by @xmath467 , and , instead of @xmath527 itself , ( ii ) the ( orthogonal ) distance of @xmath465 from @xmath472 ( or , equivalently , from @xmath140 ) , i.e. , @xmath530 ( @xmath531 ) , which we denote by @xmath474 , see figure  [ checkmap ] ; in brief , @xmath532 . in the following we investigate the behavior of the function @xmath533 under variations of the two arguments . \(i ) @xmath534 . keep @xmath535 fixed ( and assume that @xmath535 lies in the interval @xmath536 so that its image under the kasner map is finite ) . then @xmath537 is a function of the distance @xmath538 such that @xmath539 as @xmath540 ( which is a simple consequence of the regularity of the dynamical system and continuous dependence on initial data ) . it is reasonable to conjecture that the relation is in fact monotone . \(ii ) @xmath541 . keep the distance @xmath538 fixed , so that @xmath537 is a function of @xmath535 . the fundamental observation is that @xmath537 becomes unbounded as ( a ) @xmath542 , and ( b ) @xmath543 . case ( a ) is due to the intricacies of the flow in the vicinity of the non - transversally hyperbolic taub points ( where @xmath484 ) ; recall that @xmath544 measures the angular distance of the check point from the nearest taub point . for ( b ) we consider the orbit @xmath545 ( for some @xmath154 ) which corresponds to @xmath546 . suppose that @xmath472 coincides with @xmath106 ( i.e. , @xmath547 ) . in general , @xmath473 will not coincide with @xmath109 ( independently of the choice of @xmath548 ) ; therefore @xmath549 . if @xmath472 is very close to @xmath106 , which means that @xmath467 is close to @xmath194 , a small deviation of @xmath433 from the type  ii transition emanating from @xmath472 will still be small at the end point @xmath471 ; however , in the vicinity of the taub point @xmath109 even a small deviation can translate to a large error @xmath550 between @xmath469 and @xmath551 , which in turn results in the asserted blow - up of the order of magnitude of the error . the qualitative properties of @xmath537 as a function of @xmath538 and @xmath535 are depicted in figure  [ uerror ] . [ cc][cc][0.7][0]@xmath552 [ cc][cc]@xmath535 [ cc][cc]@xmath537 [ cc][cc][1][0]@xmath538    consider a generic type  ix orbit @xmath433 and its representation as an approximate sequence of transitions . by theorem  [ rinthm ] each orbit converges to the mixmaster attractor . as a consequence the distance @xmath553 ( @xmath554 ) converges to zero as @xmath459 . therefore , as @xmath555 ( @xmath556 ) , the sequence of functions @xmath557 of figure  [ uerror ] converges to zero . however , this convergence is merely pointwise and not uniform ; this behavior is crucial to understand the behavior of @xmath470 as @xmath459 . if a type  ix orbit @xmath433 converges to one of the heteroclinic cycles on the mixmaster attractor , then there exists @xmath558 such that @xmath559 for all sufficiently large @xmath256 . on this interval the function @xmath537 of figure  [ uerror ] converges to zero uniformly . we therefore obtain that @xmath560 and thus @xmath479 as @xmath459 for this type  ix orbit . however , the mixmaster attractor conjecture suggests that almost every type  ix orbit @xmath433 has an associated approximate kasner sequence @xmath468 that is unbounded , hence the general case is not so clear . since @xmath468 enters the intervals @xmath561 and @xmath562 for any @xmath558 , @xmath563 ( and thus @xmath470 ) need not necessarily converge to zero . let us elaborate . let @xmath564 be fixed . by the implicit function theorem , the inequality @xmath565 is satisfied if and only if @xmath566 or @xmath567 for some functions @xmath568 , @xmath569 , which satisfy @xmath570 and @xmath571 as @xmath572 . accordingly , for a given type  ix orbit ( and its associated approximated kasner sequence @xmath467 ) , we obtain @xmath573 or @xmath574 . we call the union of the two intervals @xmath575 and @xmath576 the ` hazard zone ' ( associated with epoch number @xmath256 ) ; obviously , the hazard zone is decreasing with @xmath256 . hence , the order of magnitude of the error , @xmath528 , is larger than @xmath577 at epoch number @xmath256 if and only if the approximate kasner parameter @xmath467 falls into the hazard zone ( associated with @xmath256 ) .    in connection with the kasner map convergence conjecture the question is how often @xmath468 enters the hazard zone , i.e. , how often   occurs : finitely many times or infinitely many times ? suppose that the mixmaster stochasticity conjecture is correct . then almost every sequence @xmath468 admits a probabilistic interpretation in terms of the probability distribution  . accordingly , we expect the question to turn into an example of a ` 01 law ' : there exists two alternatives : ( i ) type  ix orbits ( i.e. , sequences @xmath468 ) that satisfy   infinitely many times are generic ( i.e. , of measure  @xmath194 in the state space ) ; type  ix orbits that do not are non - generic ( i.e. , of measure  @xmath578 ) ; ( ii ) type  ix orbits that satisfy   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 depends on the rate of decay of @xmath579 and the decay and growth of @xmath568 and @xmath569 , respectively , as @xmath572 . case ( i ) is associated with small ( subcritical ) rates of decay ; case ( ii ) with large ( overcritical ) decay . we conjecture that case ( ii ) applies , i.e. ,   is satisfied only finitely many times for generic type  ix orbits . the reason for this conjecture is the fast rate of convergence of the orbit to the mixmaster attractor which follows from the decay of the functions @xmath580 , cf .  , and  @xmath581 , cf .  @xcite ; this leads to the expectation that the r.h . side of   represents a rapidly decaying function . to conclude our line of arguments in favor of the kasner map convergence conjecture , we note that the actual error @xmath470 can be estimated by @xmath528 therefore , if @xmath582 for all @xmath256 except for a finite set , then also @xmath583 for all @xmath256 except for a finite set . since @xmath577 is an arbitrary positive number , the statement of the kasner map convergence conjecture ensues . we round off this section with some further remarks on the conjectures . first , we note that one might be led to suppose that there could in fact exist type  ix solutions for which the statement of the kasner map convergence conjecture is violated , i.e. , @xmath480 as @xmath462 . in any case , the class of these special solutions is at most of measure zero . second , we note that the statement of the kasner map convergence conjecture , i.e. , @xmath479 ( @xmath459 ) , and the statement @xmath519 ( @xmath507 ) , cf .  the discussion on piecewise exact sequences , are expressions of one and the same stochastic 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 be in fact ` every ' . in that case , the non - generic examples of kasner / mixmaster sequences , see section  [ maps ] , would not have any counterparts among the type  ix solutions . of particular interest in this context would be to have an answer to question ( ii ) of the second subsection of section  [ beliefs ] . if there do not exist type  ix orbits whose @xmath154-limit set is one of the heteroclinic cycles , this would be a strong indication in favor of ` every ' and against ` almost every ' . if there exist type  ix orbits that converge to a cycle , ` almost every ' is the best one can aim for in the mixmaster stochasticity conjecture .    finally , let us briefly comment on claims of stochastic and chaotic properties of bianchi type  ix asymptotic dynamics . chaotic aspects of the kasner map and related maps have been studied under various aspects  @xcite . however , the relevance of these maps for type  ix asymptotics 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 of the results on the kasner map carry over to full type  ix dynamics . numerical investigations , see  @xcite and references therein , reflect theorem  [ rinthm ] and finite shadowing , but it is implausible that 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 type of stochasticity the simulation is looking for . accordingly , numerical studies will necessarily find type  ix solutions to exhibit the same stochastic behavior as generic kasner sequences and are thus not suited to explore the validity of the conjectures . in this section we briefly review the hamiltonian billiard approach to type  ix dynamics in the spirit of  @xcite and make contact with the dynamical systems approach . the metric   can be written as @xmath584 where @xmath585 is an arbitrary time variable that is related to proper time @xmath57 by a densitized lapse function @xmath586 according to @xmath587 ; evidently , @xmath588 $ ] . the hamiltonian for the orthogonal bianchi type ix perfect fluid models is given by @xmath589 where @xmath590 is three - curvature and @xmath591 , cf . the remark following   ( where we recall that @xmath592 ) . @xmath593 is the inverse of the so - called minisuperspace metric @xmath594 , @xmath595 i.e. , @xmath596 is a @xmath597-dimensional lorentzian metric . the gravitational potential @xmath598 is given by @xmath599 where we have set the structure constants @xmath600 to one ; the potential for the fluid is given by @xmath601 $ ] . for further details see , e.g. ,  @xcite or  ( * ? ? ? * chapter 10 ) ; note that @xmath602 , which is used in these references ; see also  @xcite and @xcite . as argued in  @xcite , @xmath603 is expected to be timelike in the asymptotic regime , i.e. , @xmath604 . assuming that @xmath603 is timelike allows to introduce new metric variables instead of @xmath605 . defining @xmath606 and orthogonal angular metric variables , collectively denoted by @xmath153 , leads to @xmath607 where @xmath608 is the standard metric on hyperbolic space . making a further change of variables , @xmath609 yields @xmath610\:.\ ] ] choosing the lapse according to @xmath611 leads to a hamiltonian of the form @xmath612 + \bar{\rho}^2 \sum_a c_a\,\exp(-2 \bar{\rho}\ , w_a(\gamma))\:,\ ] ] where @xmath613 denotes certain linear forms of the variables @xmath614 , i.e. , @xmath615 ; see  @xcite for details . the essential point is that one expects that @xmath616 towards the singularity and hence that each term @xmath617 $ ] becomes an infinitely high sharp wall described by an infinite step function @xmath618 that vanishes for @xmath619 and is infinite for @xmath620 . accordingly , only ` dominant ' terms in the potential are assumed to be of importance for the generic asymptotic dynamics , while ` subdominant ' terms , i.e. , terms whose exponential functions can be obtained by multiplying dominant wall terms , are neglected . in the present case there are three dominant terms in @xmath621 ( which is the minimal set of terms required to define the billiard table ) , the three exponentials @xmath622 . dropping the subdominant terms in the limit @xmath623 leads to an asymptotic hamiltonian of the form @xmath624 + \sum_{a=1}^3 \theta_\infty(-2 w_a(\gamma))\:,\ ] ] where only the three dominant terms appear in the sum . the correspondence between the dynamical systems picture and the hamiltonian picture is easily obtained by noting that the hamiltonian constraint   is proportional to the gauss constraint  . the dominant terms correspond to the terms @xmath625 , @xmath626 , in  ; the subdominant terms are collected in  @xmath400 . the non - trivial dynamics described by   resides in the variables @xmath153 , i.e. , in the hyperbolic space . it can be described asymptotically as geodesic motion in hyperbolic space constrained by the existence of sharp reflective walls , i.e. , the asymptotic dynamics is determined by the type ix ` billiard ' given in figure  [ billiardix ] . based on the heuristic considerations the limiting hamiltonian   is believed to describe generic asymptotic dynamics . [ cc][cc]@xmath627 [ cc][cc]@xmath628 [ cc][cc]@xmath629     the hamiltonian picture ( as represented by the billiard ) and the dynamical systems picture ( as represented by the mixmaster attractor ) are ` dual ' to each other . in the hamiltonian picture the asymptotic dynamics is described as the evolution of a point in the billiard . straight lines ( geodesics ) in hyperbolic space correspond to kasner states . wall bounces correspond to bianchi type  ii solutions ; the bounces change the kasner states according to the kasner map . since the billiard picture emphasizes the dynamics of the configuration space variables , one may say that the hamiltonian billiard approach yields a ` configuration space ' representations of the essential asymptotic dynamics .    in the dynamical systems state space picture the motion in the hamiltonian billiard picture becomes a ` wall ' of fixed points the kasner circle @xmath101 . the walls in the hamiltonian billiard are 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 wall bounces : the kasner map . since the variables @xmath630 are proportional to @xmath631 , it is natural to refer to the projected dynamical systems picture as a ` momentum space ' representation of the asymptotic dynamics .    summing up , ` walls ' and ` straight line motion ' switch places between the hamiltonian formulation and dynamical systems description of asymptotic dynamics , and the two pictures give equivalent complementary asymptotic pictures . the above heuristic derivation of the limiting hamiltonian rests on two basic assumptions : it is assumed that @xmath603 is timelike in the asymptotic regime and that one can drop the subdominant terms . these assumptions correspond to assuming that @xmath44 and @xmath400 can be set to zero asymptotically , i.e. , the procedure precisely assumes theorem  [ rinthm ] . ( the hamiltonian analysis in  ( * ? ? ? * chapter 2 ) uses the same assumptions , and hence the present discussion is of direct relevance for that work as well . ) that these assumptions are highly non - trivial is indicated by the difficulties that the proof of theorem  [ rinthm ] has presented ; cf . an alarming example is bianchi type  viii ; in this case the heuristic procedure in this respect leads to exactly the same asymptotic results , but so far there exist no proof that @xmath44 and @xmath400 tend to zero towards the singularity for generic solutions . we elaborate on the differences between type  viii and type  ix in  @xcite . moreover , like in the state space picture there exists no proof that all of the possible 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 systems picture and the hamiltonian billiard picture , and it is not excluded a priori that solutions are forced to one , or several , of these . we emphasize again that all proposed measures of chaos that take billiards as the starting point , see  @xcite and references therein , rely the conjectured connection between the mixmaster map and asymptotic type  ix dynamics . the above discussion shows that there are non - trivial assumptions and subtle phenomena that are being glossed over in heuristic billiard ` derivations . ' nevertheless , we do believe that the billiard procedure elegantly uncovers the main generic asymptotic features , and it might be fruitful to attempt to combine the dynamical systems and hamiltonian picture in order to prove the mixmaster conjectures . a tantalizing hint is that in the billiard picture @xmath632 becomes an asymptotic constant of the motion . rewriting this dimensional constant of the motion in terms of the dynamical systems variables yields @xmath633\:,\ ] ] where @xmath634 and @xmath635 is a constant . the purpose of this paper is two - fold . on the one hand , we analyze the main known results on the asymptotic dynamics of bianchi type  ix vacuum and orthogonal perfect fluid models towards the initial singularity . the setting for our discussion is the hubble - normalized dynamical systems approach , since this is essentially the set - up which has led to the first rigorous statements on bianchi type  ix asymptotics  @xcite . ( we choose slightly different variables to emphasize the permutation symmetry that underlies the problem . ) the main result ( ` mixmaster fact ' ) is theorem  [ rinthm ] which is due to ringstrm  @xcite ; for an alternative proof see  @xcite . this theorem , in conjunction with an analysis of the mixmaster attractor , leads to a number of further rigorous results which we list as consequences  @xcite . on the other hand , we draw a clear line between rigorous results ( ` facts ' ) and heuristic considerations ( ` beliefs ' ) . we make explicit that the implications of theorem  [ rinthm ] and its consequences are rather limited , in particular , the rigorous results do not give any information on the details of the oscillatory nature of mixmaster asymptotics . the mathematical methods required to obtain proofs about the actual asymptotic mixmaster oscillations are yet to be developed ; it is likely that radically new ideas are needed . this paper provides the infrastructure that might yield the basis for further developments . our framework enables us to transform vague beliefs to a number of specific conjectures that describe the expected ` complete picture ' of bianchi type  ix asymptotics . the arguments we give in their favor are based on an in - depth analysis of the mixmaster map and its stochastic aspects in combination with the dynamical systems concept of shadowing . we conclude with a few pertinent comments . first , as elaborated in  @xcite there exists no corresponding theorem to theorem  [ rinthm ] in other oscillatory bianchi models such as bianchi type vi@xmath636 or viii ; this suggests that the situation in the general inhomogeneous case is even more complicated than expected . furthermore , numerical studies are incapable of shedding light on the asymptotic limit . this is mainly due to the accumulation of inevitable random numerical errors that make it a priori impossible to track a particular type  ix orbit . finally , although the hamiltonian methods are a formidable heuristic tool , so far this approach has not yielded any proofs about asymptotics . nevertheless , it might prove to be beneficial to explore the possible synergies between dynamical systems and hamiltonian methods .    in this paper and in  @xcite we have encountered remarkable subtleties as regards the asymptotic dynamics of oscillatory singularities ; this emphasizes the importance of a clear distinction between facts and beliefs . we thank alan rendall , hans ringstrm , and in particular lars andersson for useful discussions . we gratefully acknowledge the hospitality of the mittag - leffler institute , where part of this work was completed . cu is supported by the swedish research council . motter and p.s . mixmaster chaos . 127 ( 2001 ) . more qualitative cosmology 137 ( 1971 ) . o.i . bogoyavlensky and s.p . novikov singularities of the cosmological model of the bianchi type ix type according to the qualitative theory of differential equations 747 ( 1973 ) . o.i . bogoyavlensky . 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( north holland , amsterdam , 1987 ) and in _ cosmology of the early universe _ , r. ruffini , l.z . fang , eds . ( world scientific , singapore , 1984 ) . arxiv : gr - qc/0102035 . collins and j.m . qualitative cosmology . 419 ( 1971 ) . lim , c. uggla , and j. wainwright . asymptotic silence - breaking singularities . 2607 ( 2006 ) . j. wainwright , m. j. hancock , and c. uggla . asymptotic self - similarity breaking at late times in cosmology . 2577 ( 1999 ) .
we consider the dynamics towards the initial singularity of bianchi type  ix vacuum and orthogonal perfect fluid models with a linear equation of state . 
 surprisingly few facts are known about the ` mixmaster ' dynamics of these models , while at the same time most of the commonly held beliefs are rather vague . in this paper 
 , we use mixmaster facts as a base to build an infrastructure that makes it possible to sharpen the main mixmaster beliefs . 
 we formulate explicit conjectures concerning ( i ) the past asymptotic states of type  ix solutions and ( ii ) the relevance of the mixmaster / kasner map for generic past asymptotic dynamics . 
 the evidence for the conjectures is based on a study of the stochastic properties of this map in conjunction with dynamical systems techniques . 
 we use a dynamical systems formulation , since this approach has so far been the only successful path to obtain theorems , but we also make comparisons with the ` metric ' and hamiltonian ` billiard ' approaches .