Randomness on full shift spaces

We give various characterizations for algorithmically random configurations on full shift spaces, based on randomness tests. We show that all nonsurjective cellular automata destroy randomness and surjective cellular automata preserve randomness. Furthermore all one-dimensional cellular automata preserve nonrandomness. The last three assertions are also true if one replaces randomness by richness – a form of pseudorandomness, which is compatible with computability. The last assertion is true even for an arbitrary dimension.