  Extending to all probability measures the notion of m-equicontinuous cellular
automata introduced for Bernoulli measures by Gilman, we show that the entropy
is null if m is an invariant measure and that the sequence of image measures of
a shift ergodic measure by iterations of such automata converges in Cesaro mean
to an invariant measure mc. Moreover this cellular automaton is still
mc-equicontinuous and the set of periodic points is dense in the topological
support of the measure mc. The last property is also true when m is invariant
and shift ergodic.
