  In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with
algebraic local rules and such that $\mathfrak{G}$ is a topological Markov
chain which has a structure compatible to this local rule. We characterize such
cellular automata and study the convergence of the Ces\`aro mean distribution
of the iterates of any probability measure with complete connections and
summable decay.
