  Revisiting the notion of m-almost equicontinuous cellular automata introduced
by R. Gilman, we show that the sequence of image measures of a shift ergodic
measure m by iterations of a m-almost equicontinuous automata F, converges in
Cesaro mean to an invariant measure mc. If the initial measure m is a
Bernouilli measure, we prove that the Cesaro mean limit measure mc is shift
mixing. Therefore we also show that for any shift ergodic and F-invariant
measure m, the existence of m-almost equicontinuous points implies that the set
of periodic points is dense in the topological support S(m) of the invariant
measure m. Finally we give a non trivial example of a couple (m-equicontinuous
cellular automata F, shift ergodic and F-invariant measure m) which has no
equicontinuous point in S(m).
