Temporal Measures for Infinite Lattices

The entropies and dimensions introduced above were defined for purely spatial sequences of site values at given times as such, they can be used to characterize sets of CA configurations. An alternative behavioral characterization is of the purely temporal sequence of site-values at a given lattice site. 

Entropies We rewrite the definitions for S f B,t) and SmJa.s B,t) in terms of length-T blocks of site-values at T successive time-steps  

Dimensions The analogous temporal set and measure dimensions, may likewise be defined  

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on 

Because of the generally unpredictable behavior of class c4 rules, dimensions cannot meaningfully be defined for rules in this class. 

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