Graphical limit sets for general cellular automata

The existing theory of graphical limit sets for cellular automata relies on algebraic structures and applies only to certain classes of cellular automata that possess this structure. We extend this theory to general cellular automata using topological methods. The starting point is the observation that the rescaled space-time diagrams, intersected with an appropriately chosen compact set, form sequences in a compact, metric space. They necessarily possess converging subsequences. In the present paper we define graphical limit sets as the collection of the accumulation points. The main result is that for a large class of cellular automata the graphical limit set defined in this way carries a group structure, which is either the trivial group consisting of one element only, or is homeomorphic to S1S1. The well known self-similar, graphical limit sets are representatives of the second class.