A note on universality in multidimensional symbolic dynamics

We show that in the category of effective ℤ-dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for d ≥ 3 there exist d-dimensional shifts of finite type which are universal for 1-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective Zd-system for d ≥ 2, and in particular SFTs cannot be universal for subactions of rank ≥ 2. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.