Limit sets of cellular automata associated to probability measures

We introduce the concept of limit set associated to a cellular automaton (C A) and shift invariant probability measure. This is a subshift whose forbid den blocks are exactly those, whose probabilities tend to zero as time tend s to infinity. We compare this probabilistic concept of limit set with the concepts of attractors, both in topological and measure-theoretic sense. We also compare this notion with that of topological limit set in different d ynamical situations.