Automaticity of coarse-graining invariant orbits of one-dimensional linearcellular automata

We consider one-dimensional linear cellular automata whose states are the i ntegers module a prime power p(d) and their orbital patterns. We are partic ulary interested in initial states and their orbital patterns which are inv ariant under a certain coarse-graining operation. We show that these coarse -graining invariant initial states are p-automatic. The relationship betwee n the solutions of a certain family of coarse-graining invariant problems c oncerning linear cellular automata over the integers module p(n) is investi gated.