On the induction operation for shift subspaces and cellular automata as presentations of dynamical systems

Consider the space of configurations from a finitely generated group to a finite alphabet. We look at the translation-invariant closed subsets of this space, and at their continuous transformations that commute with translations. It is well-known that such objects can be described “locally” via finite patterns and finitary functions; we are interested in re-using these descriptions with larger groups, a process that usually does not lead to objects isomorphic to the original ones. We first characterize, in terms of group actions, those dynamics that can be presented via structures like those above. We then prove that some properties of the “induced” entities can be deduced from those of the original ones, and vice versa. We finally show how to simulate the smaller structure into the larger one. Special attention is given to the class of sofic shifts.