asymptotic data at the exceptional spatial points are not consistent with the inequalities we assume and lead to so-called “spikes". This picture has been given a firm basis in a scalar field model with symmetries [@RW] but the status of the spikes in a general context remains unclear.

Finally, since we only deal with spacelike singularities, the classes of solutions we consider do not contain all solutions governed by the action (\[001\]). Other types of singularities ([*e.g.*]{} timelike or null ones) are known to exist. Whether these other types of singularities are general is, however, an open question.

Billiard picture
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At each spatial point, the solution of the coupled Einstein-matter system can be pictured, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space [@Chitre; @Misnerb; @tDmH; @melni]. Hyperbolic billiards are chaotic when they have finite volume and non chaotic otherwise. In this latter case, the “billiard ball" generically escapes freely to infinity after a finite number of collisions with the bounding walls. Subcritical Einstein-matter systems define infinite-volume billiards. The velocity-dominated solutions correspond precisely to the last (as $t \rightarrow 0$) free motion (after all collisions have taken place), in which the 