Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCAs) on rooted d-regular trees Td. In a first result we extend the results of \cite{pca} on trees, namely we prove that to every stationary measure ν of the PCA we can associate a space-time Gibbs measure μν on Z×Td. 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∈{1,2,3} and characterizing the invariant Bernoulli product measures.