A large part of the study of cellular automata dynamics consists in comparing pairs of configurations and their orbits. Here we focus on pairs of configurations having a finite number of differences (diamonds) like in the celebrated Garden-of-Eden theorem. First, we show that it allows to study expansive-like phenomena where classical notions of chaotic behavior like positive expansivity or strong transitivity don’t apply (in reversible CAs for instance). Second, we establish that for a large class of linear CA, diffusion of diamonds is equivalent to randomization (a large class of probability measures converge weakly towards the uniform measure under the action of the CA). Both properties are also equivalent to the absence of gliders in the CA. Finally, we give examples of reversible linear CAs that are strong randomizers (the convergence towards the uniform measure is simple and not only in Cesaro mean). This strong behavior is however provably impossible with linear CA having commuting coefficients (e.g. linear CA over cyclic groups).
['Guillaume Theyssier']
Propagation, Diffusion and Randomization in Cellular Automata