{"paper":{"title":"Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"nlin.CG","authors_text":"Bruno Kimura, Cristian Spitoni, Wioletta Ruszel","submitted_at":"2017-09-29T20:37:30Z","abstract_excerpt":"In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\\mathbb{T}^d$.\n  In a first result we extend the results of [10] 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\n  $\\mathbb{Z} \\times \\mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true.\n  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 th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00084","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}