On Nilpotency and Asymptotic Nilpotency of Cellular Automata

One of the most interesting aspects in the theory of cellular automata is the study of different types of nilpotency, that is, different ways in which a cellular automaton can force a particular symbol (usually called 0) to appear frequently in all its spacetime diagrams. The simplest such notion, called simply ‘nilpotency’, is that the cellular automaton c maps every configuration to a uniform configuration . . .000 . . ., on which it behaves as the identity, in a uniformly bounded number of steps, that is, cn is a constant map for some n. The notion of ‘weak nilpotency’, where for all x∈X , we have cn(x)= . . .000 . . . for some n, is equivalent to nilpotency in all subshifts X ⊂ SZ . We don’t know a published reference for this, and give a proof in Proposition 1, slightly strengthening Proposition 2 of [4]. There are at least two ways to define variants of nilpotency using measure theory. A notion called ‘ergodicity’ is discussed in [1], where it is also shown to be equivalent to nilpotency, while another notion called ‘unique ergodicity’ is shown to be strictly weaker in [12]. Unique ergodicity refers to the existence of a unique invariant measure for the cellular automaton, and ergodicity refers to unique ergodicity with the additional assumption that every measure converges weakly to the unique invariant measure in the orbit of the cellular automaton. A particularly nice variant of nilpotency is the ‘asymptotic nilpotency’ defined and investigated in at least [5] (where asymptotically nilpotent cellular automata are called ‘cellular automata with a nilpotent trace’) and [4]. This notion is shown to be equivalent to nilpotency on one-dimensional full shifts in [5], and on one-dimensional transitive SFTs in [4], but interestingly the proofs are not trivial, unlike for most of the notions listed above which coincide with nilpotency. It was left open in [5] whether asymptotic nilpotency implies nilpotency in all dimensions, that is, on the groups Zd for arbitrary d, and the proof given in [5] does not generalize above d = 1 as such. The problem is restated as open in [4]. In this article, we prove that asymptotic nilpotency indeed implies nilpotency in all dimensions in Theorem 2 by reducing to the one-dimensional case, and ask whether this holds in all groups. The study of nilpotency properties of cellular automata belongs to the more general study of asymptotic behavior of cellular automata, where emphasis is usually put on the limit set of a cellular automaton, see [8] for a survey on this topic. More generally, the study of cellular automata is subsumed by the study