An example of degenerate hyperbolicity in a cellular automaton with 3 states

Abstract

We show that a behaviour analogous to degenerate hyperbolicity can occur in nearest-neighbour cellular automata (CA) with three states. We construct a 3-state rule by "lifting" elementary CA rule 140. Such "lifted" rule is equivalent to rule 140 when arguments are restricted to two symbols, otherwise it behaves as identity. We analyze the structure of multi-step preimages of 0, 1 and 2 under this rule by using minimal finite state machines (FSM), and exploit regularities found in these FSM. This allows to construct explicit expressions for densities of 0s and 1s after $n$ iterations of the rule starting from Bernoulli distribution. When the initial Bernoulli distribution is symmetric, the densities of all three symbols converge to their stationary values in linearly-exponential fashion, similarly as in finite-dimensional dynamical systems with hyperbolic fixed point with degenerate eigenvalues.