We analyze the dynamics of the mixmaster universe on the basis of a standard Arnowitt-Deser-Misner Hamiltonian approach showing how its asymptotic evolution to the cosmological singularity is isomorphic to a billiard ball on the Lobachevsky plane. The key result of our study consists in the temporary gauge invariance of the billiard ball representation, once provided the use of very general Misner-Chitr\'e-like variables.
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Imponente et al. (2001) studied this question.
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