Spatio-temporal measures for infinite lattices

Entropies The individual spac e and time measures introduced above may also be generalized to space-time blocks of size B x T  

The values of all of the sites contained within the B x T) dotted inner region of this figure are completely determined by the values in the outer lined region. Since this surrounding area contains B- -2r T — 1)) sites (where V is, as usual, the range of the given CA rule), we have that 

In particular, the last relation implies that if B,T — oo with T/B fixed, then the information per site in a B x T space-time region, Iq = 5meas(B, T)/B — 0. In other words, completely random space-time patterns for one-dimensional CA can never really be generated. 

If T — oo with B fixed, the contribution by the length-B horizontal segment of the outer lined area in figure 4.15 can be effectively ignored. Noting that the entropy is largest when the 2r vertical columns are statistically independent, we find that 

Dimensions The limits of the set and measure entropies as T, B — oo yield the spatio-temporal set and measure dimensions, respectively  

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