{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:BNTW4AXZQSYA4SW6G5H6FM6RAZ","short_pith_number":"pith:BNTW4AXZ","schema_version":"1.0","canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","source":{"kind":"arxiv","id":"1309.6539","version":4},"attestation_state":"computed","paper":{"title":"On the ergodicity of geodesic flows on surfaces of nonpositive curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Weisheng Wu","submitted_at":"2013-09-25T15:03:30Z","abstract_excerpt":"Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed \"flat\" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed \"flat\" geodesics."},"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":"1309.6539","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","cross_cats_sorted":[],"title_canon_sha256":"2879908e7cdd1f0b3368b186178964e3e30dcf0970de47f1bd798c0b40d111a8","abstract_canon_sha256":"acd801754b1d4f3389b1ee79c2f6d670b7f8c1f8b4d566892a92de5bbefc6f11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:00.419834Z","signature_b64":"5d/qe4ORGy1sk/72o1rQC7McrEOgOE6ywuwtcws8HatJYNp4pykWe5hcmfFJPmKsP86hRZMrm3lw092XmRxpBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","last_reissued_at":"2026-05-18T02:20:00.419142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:00.419142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the ergodicity of geodesic flows on surfaces of nonpositive curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Weisheng Wu","submitted_at":"2013-09-25T15:03:30Z","abstract_excerpt":"Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed \"flat\" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed \"flat\" geodesics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6539","kind":"arxiv","version":4},"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":"1309.6539","created_at":"2026-05-18T02:20:00.419245+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6539v4","created_at":"2026-05-18T02:20:00.419245+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6539","created_at":"2026-05-18T02:20:00.419245+00:00"},{"alias_kind":"pith_short_12","alias_value":"BNTW4AXZQSYA","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"BNTW4AXZQSYA4SW6","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"BNTW4AXZ","created_at":"2026-05-18T12:27:40.988391+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/BNTW4AXZQSYA4SW6G5H6FM6RAZ","json":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ.json","graph_json":"https://pith.science/api/pith-number/BNTW4AXZQSYA4SW6G5H6FM6RAZ/graph.json","events_json":"https://pith.science/api/pith-number/BNTW4AXZQSYA4SW6G5H6FM6RAZ/events.json","paper":"https://pith.science/paper/BNTW4AXZ"},"agent_actions":{"view_html":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ","download_json":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ.json","view_paper":"https://pith.science/paper/BNTW4AXZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6539&json=true","fetch_graph":"https://pith.science/api/pith-number/BNTW4AXZQSYA4SW6G5H6FM6RAZ/graph.json","fetch_events":"https://pith.science/api/pith-number/BNTW4AXZQSYA4SW6G5H6FM6RAZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/action/storage_attestation","attest_author":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/action/author_attestation","sign_citation":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/action/citation_signature","submit_replication":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/action/replication_record"}},"created_at":"2026-05-18T02:20:00.419245+00:00","updated_at":"2026-05-18T02:20:00.419245+00:00"}