Covariant feature of the Mixmaster model invariant measure

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. Asymptotically to the cosmological singularity, we construct the appropriate invariant measure for the system (which relies on the appearance of an "energy-like" constant of motion) in such a way that its existence is independent of fixing the time gauge, i.e. the corresponding lapse function. The key point in our analysis consists of introducing generic Misner-Chitre-like variables containing an arbitrary function, whose specification allows us to set up the same statistical scheme in any time gauge.