Chaos and singularities in the mixmaster universe

The origin of chaotic behavior in the mixmaster model of the universe is studied from the nonlinear dynamical approach. The motion in the mixmaster model of the universe is described well by the geodesic flow in a nearly-flat Riemannian space with some rigid reflective walls. As a result, the mixmaster equations can be reduced to several maps with discrete time which stands for a sequence of Kasner periods. We construct the maps in velocity space and illustrate that the maps successfully reproduce the scaling relations obtained by numerical simulations of the mixmaster differential equations. Furthermore, the reduced maps enable us to derive new scaling laws related to the statistical behavior of the mixmaster model. It is a significant point that the mixmaster model has two types of singularities, and that they endow this model with remarkable coherence and randomness not only in the asymptotic limit toward the big crunch, but also in the asymptotic approach to the climax where the mixmaster universe takes a maximum volume. The ergodic-theoretical structure near these singular points is analyzed by means of the invariant measure which is exactly derived from the reduced map. The relation between our theory and previous ones is briefly discussed. We emphasize that the non-stationarity of the mixmaster universe model originates from the σ-infinity of the invariant measure. The breakdown of the "law of large numbers" is discussed on the basis of several results obtained by computer simulations. The phase space structure of the mixmaster universe model is discussed together with the generalization of our theory.