  We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochastic
properties; in the present paper we address either the original approach due to
Belinski, Khalatnikov and Lifshitz (BKL) as well as a Hamiltonian one relying
on the Arnowitt--Deser--Misner (ADM) reduction. We compare these two frameworks
and show how the BKL map is related to the geodesic flow associated with the
ADM dynamics. In particular, the link existing between the \textit{anisotropy
parameters} and the \textit{Kasner indices} is outlined.
