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# Trivial solutions

It would be surprising if the equations for self consistency had unique solutions. In fact, even for one-block equations, the quiescent rules will always have a solution for the quiescent state. Generally there is another solution (except for Wolfram's classes i and ii, which practically exclude additional solutions by definition.) If there is more than one solution it will be found that some of them are stable while others are unstable; in fact there is a whole theory surrounding the existence and properties of the fixed points of nonlinear equations.

Whatever may be the nature of the solutions of the local field theory equations in general, there are some solutions which can be foreseen. If the length of the block corresponds to the length of a cycle for a given automaton, then the members of the cycle can be assigned equal probabilities and other blocks can be assigned probability zero. We have two cases to consider---the blocks obtained by symmetry but belonging to the same phase of evolution, and the blocks belonging to different phases. All the phases of the same block will form a cyclic submatrix of the local structure matrix, while different symmetry images will generate additional diagonal blocks. In any event, a given block has exactly one counterimage within the sequence of evolution, and everything has been assigned equal probabilities.

There are some patterns of cyclic evolution which have no other ancestors than their immediate predecessors in the cycle of evolution---the still life of alternating zeroes and ones in Rule 22 for example, as well as the cycles of period seven amd eleven. Presumably they represent unstable fixed points in the parameter space of blocks. Other cycles may be the endpoint of various transients, and thus be somewhat more stable. Generally the self-consistent probabilities for blocks will not correspond to the actual densities of cells for any particular pattern of evolution, but rather will be formed from a composite of all of them.

It is also not excluded that there will be probabilities which are not self-consistent, but rather which alternate between a finite number of values. This behaviour tends to occur for rules which do not have quiescent states, so that alternation between two or more backgrounds can occur.    Next: Positive matrices Up: Probabilistic evolution matrix Previous: Estimating the number

Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx