Cycle-State Structure from Global Eigenvalue Spectrum

1 Cycle-State Structure from Global Eigenvalue Spectrum 

Given any arbitrary local range-r one-dimensional CA rule, (/ 0,1. fe — 

To see this, we first rewrite equation 5.1 in matrix form as 

Although any arbitrary labeling scheme for the global states may be used to define these matrices, the most convenient one for our purposes here is one that exploits our knowledge of the possible types of global states. In particular, we label according to the following scheme  

assign the lowest global-state numbers - 1,2. rig - to all garden-of-eden states that is, all states that can arise only as initial states of the system and can never be dynamically generated during the course of the subsequent evolution. 

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