Covariant Formulation of the Invariant Measure for the Mixmaster Dynamics

We provide a Hamiltonian analysis of the Mixmaster Universe dynamics showing the covariant nature of its chaotic behavior with respect to any choice of time variable. We construct the appropriate invariant measure for the system (which relies on the existence of an ``energy-like'' constant of motion) without fixing the time gauge, i.e. the corresponding lapse function. The key point in our analysis consists of introducing generic Misner-Chitr\'e-like variables containing an arbitrary function, whose specification allows one to set up the same dynamical scheme in any time gauge.