The topological skeleton of cellular automaton dynamics

We have developed statistical techniques to study the structure of the state-transition graphs of cellular automata with periodic boundary conditions, in the limit of large system size. We organize our results around the concept of a topological skeleton. The topological skeleton is the set of physically relevant states. Covering this skeleton is a surface, typically thin and dense, which contains the bulk of the set of states. States in the skeleton have some long histories. States on the surface, by contrast, have only short histories; they are reached only near the beginning of cellular automaton evolution. We study in detail a sequence of rules which exhibit mostly skeletal to mostly surface structure.