  Studies of a class of infinite one dimensional self-gravitating systems have
highlighted that, on the one hand, the spatial clustering which develops may
have scale invariant (fractal) properties, and, on the other, that they display
"self-similar" properties in their temporal evolution. The relevance of these
results to three dimensional cosmological simulations has remained unclear. We
show here that the measured exponents characterizing the scale-invariant
non-linear clustering are in excellent agreement with those derived from an
appropriately generalized "stable-clustering" hypothesis. Further an analysis
in terms of "halos" selected with a friend-of-friend algorithm reveals that
such structures are, statistically, virialized across the range of scales
corresponding to scale-invariance. Thus the strongly non-linear clustering in
these models is accurately described as a virialized fractal structure, very
much in line with the "clustering hierarchy" which Peebles originally envisaged
qualitatively as associated with stable clustering. If transposed to three
dimensions these results would imply, notably, that cold dark matter halos (or
even subhalos) are 1) not well modeled as smooth objects, and 2) that the
supposed "universality" of their profiles is, like apparent smoothness, an
artefact of poor numerical resolution.
