The Einstein equations are shown to exhibit formal chaotic behaviour that can be characterized by invariants of non-linear dynamics. An overview of new concept in dynamical systems theory is provided. The Mixmaster universe is studied as a dynamical system in an appropriate phase space, a Poincare return mapping is constructed for the system and a smooth invariant measure is calculated. Several dynamical invariants can then be calculated for the Mixmaster model, including its metric entropy. Various results in the metric theory of numbers are employed to calculate other aspects of the chaotic behaviour. Perturbations of the Mixmaster return mapping and the rate of approach to the equilibrium measure are also considered The Mixmaster model is shown to be a Bernoulli system and the Hamiltonian formulation of Misner used to display the connection between solutions to Einstein's equations and geodesic flows in hyperbolic Riemannian space. We describe the source of chaotic behaviour in the Mixmaster model, the classification of homogeneous solutions to the Einstein equations by reference to the presence of chaos, gravitational turbulence. universal behaviour in Einstein's equations and a possible description of Penrose's gravitational entropy.

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