Computational Complexity of Dynamical Systems: the case of Cellular Automata

Cellular Automata can be considered discrete dynamical systems and at the same time a model of parallel computation. In this paper we investigate the connections between dynamical and computational properties of Cellular Automata. We propose a classification of Cellular Automata according to the language complexities which rise from the basins of attraction of subshift attractors and investigate the intersection classes between our classification and other three topological classifications of Cellular Automata. From the intersection classes we can derive necessary topological properties for a cellular automaton to be computationally universal.