On the Bernoulli Automorphism of Reversible Linear Cellular Automata

TL;DR

This paper proves that reversible linear cellular automata over rac{Z}_mrac{Z}_m are either Bernoulli automorphisms or non-ergodic, resolving an open problem for this class of automata.

Contribution

It establishes a dichotomy for reversible linear cellular automata, showing they are either Bernoulli automorphisms or non-ergodic, answering an open question in the field.

Findings

Reversible linear cellular automata are either Bernoulli automorphisms or non-ergodic.

The result confirms the conjecture for the case of reversible linear cellular automata.

Provides a classification of ergodic properties for this class of automata.

Abstract

This investigation studies the ergodic properties of reversible linear cellular automata over $\mathbb{Z}_m$ for $m \in \mathbb{N}$. We show that a reversible linear cellular automaton is either a Bernoulli automorphism or non-ergodic. This gives an affirmative answer to an open problem proposed in [Pivato, Ergodc theory of cellular automata, Encyclopedia of Complexity and Systems Science, 2009, pp.~2980-3015] for the case of reversible linear cellular automata.