Ergodicity Versus Non-Ergodicity for Probabilistic Cellular Automata on Rooted Trees

#### B. Kimura, W.M. Ruszel, C. Spitoni

2019, v.25, Issue 2, 189-216

ABSTRACT

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCAs) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of \cite{pca} on trees, namely we prove that to every stationary measure $\nu$ of the PCA we can associate a space-time Gibbs measure $\mu_{\nu}$ on $\mathbb{Z} \times \mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true.
A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $d\in \{ 1,2,3\}$ and characterizing the invariant Bernoulli product measures.

Keywords: probabilistic cellular automata, d-ary trees, ergodicity, invariant measure, space-time Gibbs measure