  The asymptotic behavior of a cellular automaton iterated on a random
configuration is well described by its limit probability measure(s). In this
paper, we characterize measures and sets of measures that can be reached as
limit points after iterating a cellular automaton on a simple initial measure,
in the same spirit as SRB measures. In addition to classical topological
constraints, we exhibit necessary computational obstructions. With an
additional hypothesis of connectivity, we show these computability conditions
are sufficient by constructing a cellular automaton realising these sets, using
auxiliary states in order to perform computations. Adapting this construction,
we obtain a similar characterization for the Ces\`aro mean convergence, a Rice
theorem on the sets of limit points, and we are able to perform computation on
the set of measures, i.e. the cellular automaton converges towards a set of
limit points that depends on the initial measure. Last, under non-surjective
hypotheses, it is possible to remove auxiliary states from the construction.

Dynamical Systems