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

TL;DR

This paper demonstrates that a form of degenerate hyperbolicity can manifest in a three-state cellular automaton, using a lifted version of rule 140, and analyzes its long-term behavior through finite state machines.

Contribution

It introduces a novel three-state cellular automaton rule derived from rule 140 and analyzes its dynamics using finite state machines, revealing hyperbolic-like convergence behaviors.

Findings

Densities of symbols converge linearly-exponentially to stationary values.

Explicit formulas for symbol densities after n iterations are derived.

Behavior mimics degenerate hyperbolicity in finite-dimensional dynamical systems.

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.