  We study the notion of limit sets of cellular automata associated with
probability measures (mu-limit sets). This notion was introduced by P. Kurka
and A. Maass. It is a refinement of the classical notion of omega-limit sets
dealing with the typical long term behavior of cellular automata. It focuses on
the words whose probability of appearance does not tend to 0 as time tends to
infinity (the persistent words). In this paper, we give a characterisation of
the persistent language for non sensible cellular automata associated with
Bernouilli measures. We also study the computational complexity of these
languages. We show that the persistent language can be non-recursive. But our
main result is that the set of quasi-nilpotent cellular automata (those with a
single configuration in their mu-limit set) is neither recursively enumerable
nor co-recursively enumerable.

Discrete Mathematics;Computational Complexity;Dynamical Systems