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 $\mathbb{Z}$, or the periodic line $\cyl{n}$, depending on 2 neighbours, admit as invariant distribution the law of a space-indexed Markov chain. Our advances concerns 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 ($\mathbb{Z}$, $\cyl{n}$, 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.