Catalan numbers and power laws in cellular automaton rule 14

Abstract

We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations $n$. Using the fact that this rule conserves the number of blocks 10 and that preimages of some other blocks exhibit patterns closely related to patterns observed in rule 184, we derive expressions for the number of $n$-step preimages of all blocks of length 3. These expressions involve Catalan numbers, and together with basic properties of iterated probability measures they allow us to to compute the density of ones after $n$ iterations, as well as probabilities of occurrence of arbitrary block of length smaller or equal to 3.