{"paper":{"title":"Liv\\v{s}ic Theorem for Banach Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Genady Ya. Grabarnik, Misha Guysinsky","submitted_at":"2014-08-24T21:56:24Z","abstract_excerpt":"We prove the Liv\\v{s}ic Theorem for H\\\"{o}lder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${\\sigma:X\\to X}$ that satisfies the Anosov Closing Lemma, and a H\\\"{o}lder continuous map ${a:X\\to B^\\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a H\\\"{o}lder continuous transition function if and only if ${a(\\sigma^{n-1}p)\\ldots a(\\sigma p)a(p)=e}$ for each periodic point $p=\\sigma^n p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5639","kind":"arxiv","version":1},"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"}