  In the turbulent regime of coupled map lattice with non-local interaction the
maps systematically form periodic cluster attractors and their remnants by
synchronization due to the foliation of periodic windows of the element map. We
examine these periodicity manifestations in three types of coupled map lattices
in $D=1, 2, 3$. In the first two, the interaction is all-to-all but the
coupling decreases with distance in a power and an exponential law. In the
third, the interaction is uniform but cut off sharply. We find that in all
three models and in all dimensions periodicity manifests universally from
turbulence when the same suppression of the local mean field fluctuation is
achieved by the non-local averaging.
