Cellular Automata and Continuous Functions: Negative Results

Let ω = {0, 1, . . . , n − 1} be a finite alphabet, DN = {1, 2, . . . ,N}, and BN = {x ∈ [0, 1] | ∃k ∈ ! : x = k/nN}. A configuration is a function of the form: ξ : DN → ω, and CN is the set of all configurations. Two configurations ξ1 and ξ2 are near if d(ξ1, ξ2) = (N −A)/N is small, where A = sup{p | ∃i ∈ {0, 1, . . . ,N} : ∀k = i + 1, i + 2, . . . , i + p ≤ N ξ1(k) = ξ2(k)}. The following results are proved. 1. There is no sequence of functions φN : CN → BN such that φN and φ−1 N uniformly converge to continuous functions in such a topology. 2. Evolutions of cellular automata (CA) cannot be approximated by the superpositions of real continuous functions. In the proofs of these results advantage was taken of some CA acting in " and in DN with a stationary boundary condition.