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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 $p=\\sigma^n p$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.5639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-24T21:56:24Z","cross_cats_sorted":[],"title_canon_sha256":"db9418e99bee9a230386f4370f0a568a3f2faaa9c4ceabcabe368491a2ed7e88","abstract_canon_sha256":"c395634ccc209c00cedb55797d7e470895298b31df434138cb94bbf970d61ede"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:24.649219Z","signature_b64":"wP5J8ZI4uhZoMFPiDgVxAAxjbYNSAGKsFZuj6h8XTiTmGFn73YqGzyPiHIrjmkQh8H6bQhCFB1ZJ6w0a0adkBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1bb5cd2618972c69dc5d3fd5a3f7592d474bcb4ea85de62e51ec58f315d081d","last_reissued_at":"2026-05-18T02:44:24.648837Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:24.648837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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