Markovianity of the invariant distribution of probabilistic cellular automata on the line

We revisit the problem of finding the conditions under which synchronous probabilistic cellular automata indexed by the line ZZ, or the periodic line Z/nZZ/nZ, depending on 2 neighbours, admit as invariant distribution the law of a space-indexed Markov chain. Our advances concern PCA defined on a finite alphabet, where most of existing results concern size 2 alphabet.A part of the paper is also devoted to the comparison of different structures (ZZ, Z/nZZ/nZ, and also some structures constituted with two consecutive lines of the space–time diagram) with respect to the property to possess a Markovian invariant distribution.