Embedding odometers in cellular automata

TL;DR

This paper investigates how odometers, a type of dynamical system, can be embedded into one-dimensional cellular automata, revealing conditions for such embeddings based on the odometer's properties.

Contribution

It demonstrates that all odometers can be embedded in gliders with reflecting walls cellular automata and characterizes when they can be embedded in group endomorphism automata.

Findings

All odometers embed in gliders with reflecting walls automata.

Finitary odometers can be embedded in group endomorphism automata.

Embedding depends on the odometer's finitary property.

Abstract

We consider the problem of embedding odometers in one-dimensional cellular automata. We show that (1) every odometer can be be embedded in a gliders with reflecting walls cellular automaton, which one depending on the odometer, and (2) an odometer can be embedded in a cellular automaton, which is a group endomorphism on an n-letter group, and where n depends on the odometer, if and only if the odometer is "finitary."