{"paper":{"title":"On universal continuous actions on the Cantor set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"G\\'abor Elek","submitted_at":"2018-03-14T18:13:27Z","abstract_excerpt":"Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\\Gamma$, there exists a free continuous action $\\zeta: \\Gamma \\curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\\alpha: \\Gamma \\curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\\Theta_\\alpha: X\\to C$ such that $\\Theta_\\alpha\\circ \\alpha=\\zeta \\circ \\Theta_\\alpha$. We extend our theorem for (nonfree) Borel $(\\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.05461","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"}