  Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description
of the asymptotic behavior of a generic solution of Einstein equations near a
spacelike singularity could be drastically simplified by considering that the
time derivatives of the metric asymptotically dominate (except at a sequence of
instants, in the `chaotic case') over the spatial derivatives. We present a
precise formulation of the BKL conjecture (in the chaotic case) that consists
of basically three elements: (i) we parametrize the spatial metric $g_{ij}$ by
means of \it{Iwasawa variables} $\beta^a, {\cal N}^a{}_i$); (ii) we define, at
each spatial point, a (chaotic) \it{asymptotic evolution system} made of
ordinary differential equations for the Iwasawa variables; and (iii) we
characterize the exact Einstein solutions $\beta, {\cal{N}}$ whose asymptotic
behavior is described by a solution $\beta_{[0]}, {\cal N}_{[0]}$ of the
previous evolution system by means of a `\it{generalized Fuchsian system}' for
the differenced variables $\bar \beta = \beta - \beta_{[0]}$, $\bar {\cal N} =
{\cal N} - {\cal N}_{[0]}$, and by requiring that $\bar \beta$ and $\bar {\cal
N}$ tend to zero on the singularity. We also show that, in spite of the
apparently chaotic infinite succession of `Kasner epochs' near the singularity,
there exists a well-defined \it{asymptotic geometrical structure} on the
singularity : it is described by a \it{partially framed flag}. Our treatment
encompasses Einstein-matter systems (comprising scalar and p-forms), and also
shows how the use of Iwasawa variables can simplify the usual (`asymptotically
velocity term dominated') description of non-chaotic systems.
