# 4.2. Mixmaster return mapping

where 7 (0) = 0 and T\ [0, l]-»[0, 1]. In analytic form this return mapping is the single-valued function.

We are now in a position to synthesize these ideas with those introduced in section 2. If one intersects the phase space of solution trajectories to the Einstein equations (4.4-4.8), which can be reduced to two real degrees of freedom, with a one-dimensional surface X which encodes all the Kasner behaviour then the trajectories recurrently intersect X to establish a Poincare return mapping which is one-dimensional. When compactified to the unit interval the Poincare return map is just given by the appropriate form of (4.14) and (4.18)

This function is shown in fig. 3, it possesses an infinite number of discontinuities and is not infective since each x0 has a countable infinity of inverse images, one on each interval [(k + I V k '] for integral k.

The values of the sequence {jc0, Tx0, T2jc0, ...} encode a Mixmaster solution by a series of nearby Kasner behaviours where the exact values of the Kasner indices can be calculated from the value of jcj. The nature of the coding is indicated in fig. 3, each iterate of the return mapping describes the Kasner behaviour initiating a new cycle of small oscillations. The return mapping is expansive, |T"(jc)| > 1 on x 6 (0 1), everywhere; in particular, all the fixed points (Tx = jc), are unstable.

In order to characterize the Mixmaster Universe as a dynamical system we must find the smooth invariant Measure preserved by T Tnis is found by the same methods as were employed in the illustrative example (2.3)-(2.9). In this case, the mapping has an infinite number of branches rather than simply two. We can establish that this measure exists and possesses strong statistical properties by using some known theorems.