{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:E4BLQW3SZ6G3ZFHSWLRWNITT6E","short_pith_number":"pith:E4BLQW3S","canonical_record":{"source":{"id":"1111.2420","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-10T08:47:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0cf96bf3d2f352dd13f41cc0ddefaeba4e86ca76569910c532adae966c3176c9","abstract_canon_sha256":"eda81f91355f9738904770b23fc5e34e2bba8563012b85fced15a8713ed3faca"},"schema_version":"1.0"},"canonical_sha256":"2702b85b72cf8dbc94f2b2e366a273f1026ce771438b7e79729016333530f0da","source":{"kind":"arxiv","id":"1111.2420","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2420","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2420v2","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2420","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"pith_short_12","alias_value":"E4BLQW3SZ6G3","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E4BLQW3SZ6G3ZFHS","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E4BLQW3S","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:E4BLQW3SZ6G3ZFHSWLRWNITT6E","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2420","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-10T08:47:09Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0cf96bf3d2f352dd13f41cc0ddefaeba4e86ca76569910c532adae966c3176c9","abstract_canon_sha256":"eda81f91355f9738904770b23fc5e34e2bba8563012b85fced15a8713ed3faca"},"schema_version":"1.0"},"canonical_sha256":"2702b85b72cf8dbc94f2b2e366a273f1026ce771438b7e79729016333530f0da","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:36.435676Z","signature_b64":"AlX9hAu1M6NlHG9QR/+A+Nd9li3z3ley3EU5LGNOCAR8KoVo30P1LhBPNIEAK6ThD5YcAQjxFtuxiCcmLlmTCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2702b85b72cf8dbc94f2b2e366a273f1026ce771438b7e79729016333530f0da","last_reissued_at":"2026-05-17T23:53:36.434871Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:36.434871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2420","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uR+FJrwmPzzQgQ2710VydL5+xZMUmnV1DTvRaBIOBcZnRT+q0ZKBs6HJkQATOwrXgzseTmPsuR/cSB1E4tqJDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T05:51:30.789035Z"},"content_sha256":"c1899d04c73eb90c3b4919675f8540de09ffd712c514b32f4f3c41f0cb213b82","schema_version":"1.0","event_id":"sha256:c1899d04c73eb90c3b4919675f8540de09ffd712c514b32f4f3c41f0cb213b82"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:E4BLQW3SZ6G3ZFHSWLRWNITT6E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Measure-theoretic chaos (Chaos au sens de la mesure)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Tomasz Downarowicz, Yves Lacroix","submitted_at":"2011-11-10T08:47:09Z","abstract_excerpt":"We define new isomorphism-invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaoses {\\rm DC2} and its slightly stronger version (which we denote by {\\rm DC}{\\small$1\\tfrac12$}). We prove that: 1. If a \\tl\\ system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is \\tl ly {\\rm DC2} ({\\rm DC}{\\small$1 \\tfrac12$}) chaotic. 2. Every ergodic system with positive Kolmogorov--Sinai entropy "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2420","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:53:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GoAekcBw/BhK7NW1/2umfOynd2pnILq9zf+PONZb/7j/Iyrk/VyHZwc7jE6DmAmSWEfhZfT2aDPQIRZBAQy3CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T05:51:30.789407Z"},"content_sha256":"03881428f444ce1e42e1ff04903957c99f0bf8045d30df9d8bb9f197b8bd0234","schema_version":"1.0","event_id":"sha256:03881428f444ce1e42e1ff04903957c99f0bf8045d30df9d8bb9f197b8bd0234"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/bundle.json","state_url":"https://pith.science/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T05:51:30Z","links":{"resolver":"https://pith.science/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E","bundle":"https://pith.science/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/bundle.json","state":"https://pith.science/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E4BLQW3SZ6G3ZFHSWLRWNITT6E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:E4BLQW3SZ6G3ZFHSWLRWNITT6E","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"eda81f91355f9738904770b23fc5e34e2bba8563012b85fced15a8713ed3faca","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-10T08:47:09Z","title_canon_sha256":"0cf96bf3d2f352dd13f41cc0ddefaeba4e86ca76569910c532adae966c3176c9"},"schema_version":"1.0","source":{"id":"1111.2420","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2420","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2420v2","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2420","created_at":"2026-05-17T23:53:36Z"},{"alias_kind":"pith_short_12","alias_value":"E4BLQW3SZ6G3","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E4BLQW3SZ6G3ZFHS","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E4BLQW3S","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:03881428f444ce1e42e1ff04903957c99f0bf8045d30df9d8bb9f197b8bd0234","target":"graph","created_at":"2026-05-17T23:53:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We define new isomorphism-invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaoses {\\rm DC2} and its slightly stronger version (which we denote by {\\rm DC}{\\small$1\\tfrac12$}). We prove that: 1. If a \\tl\\ system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is \\tl ly {\\rm DC2} ({\\rm DC}{\\small$1 \\tfrac12$}) chaotic. 2. Every ergodic system with positive Kolmogorov--Sinai entropy ","authors_text":"Tomasz Downarowicz, Yves Lacroix","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-10T08:47:09Z","title":"Measure-theoretic chaos (Chaos au sens de la mesure)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2420","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c1899d04c73eb90c3b4919675f8540de09ffd712c514b32f4f3c41f0cb213b82","target":"record","created_at":"2026-05-17T23:53:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"eda81f91355f9738904770b23fc5e34e2bba8563012b85fced15a8713ed3faca","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-10T08:47:09Z","title_canon_sha256":"0cf96bf3d2f352dd13f41cc0ddefaeba4e86ca76569910c532adae966c3176c9"},"schema_version":"1.0","source":{"id":"1111.2420","kind":"arxiv","version":2}},"canonical_sha256":"2702b85b72cf8dbc94f2b2e366a273f1026ce771438b7e79729016333530f0da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2702b85b72cf8dbc94f2b2e366a273f1026ce771438b7e79729016333530f0da","first_computed_at":"2026-05-17T23:53:36.434871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:36.434871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AlX9hAu1M6NlHG9QR/+A+Nd9li3z3ley3EU5LGNOCAR8KoVo30P1LhBPNIEAK6ThD5YcAQjxFtuxiCcmLlmTCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:36.435676Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2420","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1899d04c73eb90c3b4919675f8540de09ffd712c514b32f4f3c41f0cb213b82","sha256:03881428f444ce1e42e1ff04903957c99f0bf8045d30df9d8bb9f197b8bd0234"],"state_sha256":"87e36317d1a16c5181b48704d81ebd7e26f007c92f847734852515f3d6702c20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xpUjYb3YQNCKm9LnhcJC8nWlKAExuTlUQ1sFsajETv2jar9+i0UCKv9HZQidNHwEW64G4EXxC6CLJENGTRn3Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T05:51:30.792034Z","bundle_sha256":"06f1511f160c551e4d292688d80ee48e7a3465c07aa136f84a782bd2ffdf2e0f"}}