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We will show that if the one-sided topological Markov shifts (X_A,\\sigma_A) and (X_B,\\sigma_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\\bar X_A,\\bar\\sigma_A) and (\\bar X_B,\\bar\\sigma_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,\\sigma_A) and (X_B,\\sigma_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B)."},"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":"1307.1299","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-07-04T12:30:50Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"54bf255d5f243c7dade168c4f1960acff9d57844b356148ee93096bab1dd7a26","abstract_canon_sha256":"8713f1e4f130315dbceb7aafeaf532e56b5abc2858a104cd90d0d3499de1d636"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:15.188635Z","signature_b64":"j4hiA1ArQelkDase/WnuWO9IKi8d/k2JWVZCnzZFWCAZ6uPFv2jKyCbZTq6FUuyCZXXW/yivT9KuNLXXq7fGCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8efa7b96237441b7a3c76d6f6c0d23a8f37233d90d02cc798b2bf6e46a840d8","last_reissued_at":"2026-05-18T01:28:15.188089Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:15.188089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.OA","authors_text":"Hiroki Matui, Kengo Matsumoto","submitted_at":"2013-07-04T12:30:50Z","abstract_excerpt":"Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,\\sigma_A) and (X_B,\\sigma_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\\bar X_A,\\bar\\sigma_A) and (\\bar X_B,\\bar\\sigma_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,\\sigma_A) and (X_B,\\sigma_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1299","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1307.1299","created_at":"2026-05-18T01:28:15.188183+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.1299v1","created_at":"2026-05-18T01:28:15.188183+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1299","created_at":"2026-05-18T01:28:15.188183+00:00"},{"alias_kind":"pith_short_12","alias_value":"3DX2POLCG5CB","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"3DX2POLCG5CBW6R4","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"3DX2POLC","created_at":"2026-05-18T12:27:32.513160+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK","json":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK.json","graph_json":"https://pith.science/api/pith-number/3DX2POLCG5CBW6R4O3LPNQGSHK/graph.json","events_json":"https://pith.science/api/pith-number/3DX2POLCG5CBW6R4O3LPNQGSHK/events.json","paper":"https://pith.science/paper/3DX2POLC"},"agent_actions":{"view_html":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK","download_json":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK.json","view_paper":"https://pith.science/paper/3DX2POLC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.1299&json=true","fetch_graph":"https://pith.science/api/pith-number/3DX2POLCG5CBW6R4O3LPNQGSHK/graph.json","fetch_events":"https://pith.science/api/pith-number/3DX2POLCG5CBW6R4O3LPNQGSHK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK/action/storage_attestation","attest_author":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK/action/author_attestation","sign_citation":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK/action/citation_signature","submit_replication":"https://pith.science/pith/3DX2POLCG5CBW6R4O3LPNQGSHK/action/replication_record"}},"created_at":"2026-05-18T01:28:15.188183+00:00","updated_at":"2026-05-18T01:28:15.188183+00:00"}