Complete ergodicity in one-dimensional reversible cellular automata

TL;DR

This paper classifies and proves ergodicity for boundary-driven semi-infinite cellular automata with 3 to 5 states, providing analytical and numerical results on their ergodic rules.

Contribution

It systematically identifies all ergodic rules in CA with 3, 4, and 5 states and classifies their ergodic structures.

Findings

12 rules in 3-state CA are proven ergodic analytically

118,320 rules in 5-state CA are proven ergodic analytically

Numerical confirmation shows non-ergodicity for other rules under some boundary conditions

Abstract

Exactly ergodicity in boundary-driven semi-infinite cellular automata (CA) are investigated. We establish all the ergodic rules in CA with 3, 4, and 5 states. We analytically prove the ergodicity for 12 rules in 3-state CA and 118320 rules in 5-state CA with any ergodic and periodic boundary condition, and numerically confirm all the other rules non-ergodic with some boundary condition. We classify ergodic rules into several patterns, which exhibit a variety of ergodic structure.