4.4. Continued fractions

where the partial quotients, ifc,, k2,..., are positive integers. More concisely, we shall express the continued fraction expansion (cfe) of x ast

Consider some x f= (0. 1) and express it as a continued fraction [44,41]






The integers {ki\ can be expressed in terms of the function [;] since

tFor example, » = {3.7,15,1,292, 1,1,...} was first calculated by Wallis [46] in 1685. Continuée fractions have been used to model cosmological behaviour by George Graham. In the lS'ch century, Graham constructed pi netaria (mechanical models of the solar system), in which continued fraction expansions were used to determine the gear ratios necessa-y for the working models. Grah am was under the patronage of Charles Boyle, the 4th Earl of Orrery at this time and renamed his devices 'Orreries' in honour cf his Irish lenefactar

If x is a rational number, then T"(x) = 0 for all but a finite set of ns. Thus every rational number has a finite cfe whilst every irrational number has a unique, infinite cfe. Conversely, every infinite sequence of integers {ki} uniquely defines an irrational number with partial quotients {k,). Since we shall bt

concerned with the metric properties of continued fractions that hold for almost every x with respect to some measure we shall suppose x to be irrational with an infinite cfe.

and so on. We can simplify these representations by expressing them in terms of the transformation T{x) given by (4.20), hence k2 = kx(T(x)) and so, in general,


The representation (4.32)-<4-36) in conjunction with (4.20) display a remarkable feature of the Mixmaster Un;verse which was first recognized by Belinskii et al. (31): If the initial conditions for the Mixmaster oscillations are specified by a single irrational number, then the partial quotients in the infinite continued fraction expansion of this number correspond to the number of small oscillations within each major cycle. It is clear from our previous discussion of (4.20) that if we endow the unit interval with a tr-algebra of Lebesgue measurable sets then the k, are random variables defined almost everywhere with respect to any probability measure which assigns measure zero to the set of rational numbers and in particular with respect to Lebesgue measure.

Since the Mixmaster dynamical system (4.20) is ergodic we can apply the ergodic theorem in the form (4.28) to establish some interesting results about the average length of the Mixmaster cycles. It is not obvious that any general results of this nature should exist: Any initial ,v0 value will have a unique cfc and encode an infinite sequence of Mixmaster cycles; similarly, any infinite set of cycle lengths determine a unique initial value, jc0, because its cfe is unique. Albeit, it transpires that if we confine our attention to the continued fraction expansions of almost every jc0, or equivalently, to almost every Mixmaster evolution, then strong conclusions are possible. With this proviso in mind we can calculate the average (arithmetic mean) length of a Mixmaster cycle by evaluating the probability that the integer k appears in the cfe of a typical number. If we use the fact that T is ergodic and pick / to be the indicator function (section 2.3) of the set of {*} with ky{x) equal to r E I* then N(r), the frequency of the integer r amongst the sequence of partial quotients {£,(*)}, is by (4.28) almost everywhere equal to



Therefore, the arithmetic mean cycle length « 1 SP-t A:, does not possess a finite value. The frequency behaves as N(r)~ r 2 as r-x and so the mean approaches infinity like logr and for an infinite number of n values


Although there is no average Mixmaster cycl; length, it is instructive to evaluate the frequency of cycles of length r from (4.37), we obtain [44]



and the frequencies fall-off rapidlyt for larger r. Therefore Mixmaster cycles are most likely to be very short, over 41% of than will involve a single oscillation. This is clearly a different state of affairs to that intuitively imagined for Mixmaster evolution.

then we have

where k is a constant. Therefore


we cm find a series expansion in n


t By way of an 'experiment' if we examine the first hundred partial quotients in the cfe of rr we find excellent agreement with the predu. ion (4.39) derived from (4.37) with N,(l) = 0.41, N„(2) = 0-22, A/»(3) = 0.07 ctc. Thus, of the first one hundred cycles of a Mixmaster model codec tn in - v- [ir), forty-one have just a single oscillation.

If we choose in the ergodic theorem (4.28) a function fey1 given by






The absolute constant k was first found by Khinchin [47] and is given by (4.41) as a slowly convergent infinite product



The value of Khinchin's constant can be calculated to arbitrarily high accuracy by finding a series approximation to (4.43). By writing (4.43) in logarithmic form as








and this can be computed by using the tabulated v alues of the Riemann zeta function {(X) wheret

Numerical evaluation yields [48]$

The geometric mean cycle length for Mixmaster oscillations is given by this universrl constant of number theory.§ It confirms the evidence of the distributions (4.39): Mixmaster cycles are overwhelm­ingly likely to be short and contain 1, 2 or 3 oscillations.


and the accuracy of approximation to some irrational 6 is bounded by

To calculate a value for the topological entropy of the Mixmaster Universe we need to introduce the idea of continued fraction convergents [44,45]. Every rational possesses a continued fraction expansion of finite length. If we truncate an infinite cfe of an irrational number after a finite number of terms, r, we will obtain a rational approximation, pjq„ to the original irrational numbir; where p, qEZ*. We shall call it the ifcth convergent of the continued fraction expansion. These convergents converge very rapidly11 with r. For all expansions the denominators increase in geometric progression [44].



If we evolve a Mixmaster model whose oscillatory evolution is described by the partial quotients in the cfe of some irrational 8 then we would code the Mixmaster trajectory in phase space by following the sequence of convergents as This produces a symbolic orbit for the dynamics. Now the frequency of different symbolic orbits of length n which are separated by a distance ;s q„(q„ + <?„_,) if e is chosen to be q~l(qn +qnn)~l. Now because of the property (4.49), the limit e -»0 is equivalent to n -»« mid so the topological entropy, f/(T)„ is, by (2.39) just

+ It is also possible lo transform (4.4?) into an integral representation [48] involving the gamma function. T(x).


§ Again, as an experiment' if m calculate the geometric mean of the first hundred partial quotients of ir it seems to lie in the dense set of irrationals for which (4.42) holds with (Jti(jr) ■ ■ • W))001 = 2.6831.

1 If the cfe for w is truncated afur 1.2,3.... terms respectively we obtain the familiar rational approximants 3, 22/7, 333/106           

JJD. Bum*, Chaotic bthnkmr in genval relativity 27






This is in accord widi the uniqueness of the smooth measure and the relation (2.41).