5.1. Perturbations

However, Uvy [49] has shown that for almost every real .* the convergent denominators, qn, satisfy a universal relation:

 

and therefore the topological entropy H(T) is, using (4.52) in (4.51),

We have seen that the Mixmaster Universe is a dynamical system which displays sensitive depen­dence upon initial conditions. If we pick some space-time variable and follow its evolution through a series of Mixmaster oscillations then we wiP not be able to follow its evolution deterministically unless we know its initial state exactly. Any numerical simulation of the cosmological dynamics will \ave to face this problem: If small round-off or computational errors arise at any stage in the evolution they will eventually be amplified to such an extent that the real evolutionary behaviour is swamped by information loss just as in the simple example (1.1). However, in such sensitive dynamical systems the metric entropy is the natural variable for a description of the evolution: unlike many other parameters it responds slowly when the system is slightly perturbed and so provides a stable description of the behaviour.

where

To demonstrate this response we can consider a simple perturbation of the Mixmaster return map by a small, constant e > 0 to

 

 

A smooth invariant measure still exists for Te when e > 0 and is absolutely continuous with respect to Lebesgue measure. Solving (2.10) and using (2.32) we find the metric enlropy is just

As b increases so does the entropy. Although the modulus of |r(x)! is unaffected by the perturbation the presence of s >0 creates more chaotic behaviour because of the change in the asymptotic measure. Trajectories are now more likely to be found in the neighbourhood of very unstable points close to

v - 0 where the values of |T'(jc)| are largest.

This weak perturbation can be interpreted physically in the context of the Mixmaster evolution: The presence of h>0 corresponds to a deviation from the exact Kasner to Kasner evolution during a sequence of small oscillations within each major cycle when measured on a time interval in which there are no local maxima or minima of a2(t), hz(t) and c2(t).

Another perspective on (5.1) might arise in the context of more general inhomogeneous space-times. If we were to examine space-times which neighbour Mixmaster in the space of all solutions to Einstein's equations !", or alternatively, if we perturbed [50] the Mixmaster model, the resulting dynamics might be anticipated to possess a 'neighbouring" Poincsre return mapping, perhaps of the form (5.1).

 

where / is a C (r & 2) function (the Mixmaster map has f(x) = jc). Mappings of the form (5.5) admit the possibility of altering the derivative of the transformation as well as the asymptotic measure; both changes contribute to the metric entropy. It is known [51] that (5.5) always possesses an invariant C~'

measure, fi, satisfying

 

More realistic return maps for neighbouring space-times can be imagined: The Mixmaster map is a member of a general class of one-dimensioial maps of the form

and so the metric entropy can be calculated from (2.32), either analytically or numerically. This extension would allow us to drop the assumption that the cycle to cycle evolution of the Mixmaster is a transition from one exact Kasner model to another.

Space-times that are more general than Mixmaster and which possess spatial degrees of ireedom will not have return mappings which are one-dimensional. The Mixmaster behaviour suggests that in these more general situations the Poincare return mappings may correspond to higher-dimension;.! extensions of (5.5) called F-expansions [52]. These were first studied by Jacobi [53] and Perron [54]. If F = f~l is a continuous mapping from some convex subset of R" into (0.1)" and if we define

 

Barrov and liplcr |11) have examined the structure of this solution space in the neighbourhood of the Mixmaster model. They find that in vacuum there does not exist a ditfeomorphism from the neighbourhood of Mixmaster data onto the W* (s ? M Hilhen «pac^ of four functions of Tii;ee variable-, which paramctcn/e the solution space (although a homeomorphism does always exist). This is a specific extension of a result of hsi hci •Marsden and Moncncf |.Ui who showed that if a compact solution to the empty-space Einstein equations has Killing vectors, then the M,;u; S'-ti'e ■"> not dillcrcntiaMe at that point Roughlv speaking, the solution space has a conical structure at solutions with symmetries and an manne number ot tangents (approximations) can be drawn at the apex of this cone. Only some of these tangents will represent linearizations of other true solutions to the Einstein equations, the remainder are approximations to non-solutions. Close to Mixnaster these spurious tangents (or perturbations) arc open dense in the space of solutions to the lit earized equations and the relationship of Mixmaster to 'nearby' general solutions is liU'lv to be extremely delicate

ID. Barrow. Chaotic behaviour in general relativity

one obtains an n-dimensional analogue of the cfe known as the Jacobi algori hm.

The recurrent behaviour of the Mixmaster is suggestive of a connection between its neighbours in solution space and these n-dimensional mappings which generalize the continued fraction expansion to higher dimensions.