Statistical Mechanics of CA (Wolfram)
Using an elementary CA on a tape with 0,1
Using nearest neighbor deterministic rules
Simple initial configurations CA tend either to homogenous states, or generate self-similar patterns with fractal dimensions (1.59 ~ 1.69)
Random initial configurations tend to two universality classes, independent of the properties of the initial state or the rules.
Algebraic properties (Wolfram et. al.) include (usually) irreversibility, evolution through transients to attractors consisting of cycles sometimes containing a large number of configurations.