4.3. Invariant measure and metric entropy

T is piece-wise expanding, that is inf*|T'(x)|> I, on each monotonic interval (1 /(k + 1), 1 Ik). Now lasota and Yorke [37] have shown that if T is C2 piece-wise expanding then there must exist a smooth invariant probability measure for T that is absolutely continuous with respect to Lebesgue measure. + Furthermore, if llT'(x) is of bounded variation! then this measure is ergodic [38]. If the measure is weak mixing, then it is also a Bernoulli shift [39].

where A is a measurable set. Consider the inverse image of an open interval (a, />)£ [('. 1] under T

Now, let us endow the unit interval with the <x-a!gebra of Lebesgue measurable sets and construct this invariant probability measure n{x) which must satisfy.

 

 

 

 

 

Since this is a union of intervals which are all disjoint (4.22) reduces to solving.

1/(n + a)

l(a,b))= i I fi(x)6x.       I4 ^

J

1 /(n + b)

Now

tThat is, they havr: the same sUs of measure zero.

$ A function is of bounded variation if and only if it is the difference of two monotone functions.

 

 

In contrast to the mapping T this invariant measure is completely smooth and gives the probability of a particular Kasner configuration being visited during the Mixmaster evolution [31]. The existence of /j. characterizes the Mixmaster Universe as a measurable dynamical system.

Since T is ergodic with respect to n we can apply the ergodic theorem, (2.14), so

 

 

T also satisfies the criteria for Poincare recurrence, so for every set S of non-zero measure and for almost every .r E S there ca'* be found an infinite ?nd increasing sequence of numbers «(/) with T"0)x £ S. It follows frcrr. this that each Kasner solution is visited an arbitrarily larpe number of times during the infinite sequence of oscillations experienced by the Mixmaster Universe.

so the condition (4.22) reduces to

 

As can be readily checked the normalized solution is positive and

It can be shown that T is mixing and therefore, by the above-cited theorem, isomorphic to a Bernoulli shift. Alternatively, this isomorphism can be proved directly [40]. We shall examine some of its consequences below.

Referring back to eq. (2.32), and the theory leading up lo it, we can now calculate the metric entropy of the Mixmaster return mapping T; using (4.20) and (4.27) we have [5]

 

 

This single invariant, h(n, T) = 3.4237.... characterizes the chaotic structure of the Mixmaster Uni­verse. The Mixmaster dynamical system will be isomorphic to any Bernoulli shift possessing identical

»•trie entropy or, not« precisely if (X, f, h a Bernoulli shift with entropy ?r2/6(log 2f and (X, T, n) k Ike Mixmitxter system then there must exist a mapping       which is measure-preserving and

commutes with T and t, so

 

 

where

 

 

This analysis leads us to suggest a formal definition for the concept of a 'chaotic' cosmological model first introduced by Misner to characterize Universes that begin in a highly turbulent state but gradually evolve towards regularity as they lose memory of their initial conditions [41-43]: A chaotic cosmology is a solution of Einstein's equations possessing a non-zero metric entropy. Models with this property have the chaotic behaviour envisaged by Misner and also lose memory of their initial data after -h~] evolution times. The rate of this information loss has been interpreted in equations (2.35)-(2.37).

A closer analysis of the Mixmaster evolution reveals the origin of the non-zero dynamical entropy. It is the cycle to cycle evolution (jc-**-1) which generates the sensitive dependence on initial conditions not the small oscillations during a major cycle (x-*x - 1). Accordingly, the entropy of the Bianchi I (Kasner) and Bianchi II Universes, which describe the behaviour during smail oscillations, is zero.