A non-ergodic probabilistic cellular automaton with a unique invariant measure

TL;DR

This paper presents a probabilistic cellular automaton that uniquely has an invariant measure but is non-ergodic, providing a counterexample to a long-standing open question about the relationship between invariance and ergodicity.

Contribution

It constructs a specific PCA with a unique invariant measure that defies the typical link to ergodicity, answering an open theoretical question.

Findings

The PCA has a unique invariant measure.

The PCA is non-ergodic despite having a unique invariant measure.

This provides a counterexample to the assumption that uniqueness implies ergodicity.

Abstract

We exhibit a Probabilistic Cellular Automaton (PCA) on the integers with an alphabet and a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.