{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ZL3HH3FFYH4N3KOVYU7UJ4J23M","short_pith_number":"pith:ZL3HH3FF","schema_version":"1.0","canonical_sha256":"caf673eca5c1f8dda9d5c53f44f13adb3d686ce9d0f3604278bb392ac8264999","source":{"kind":"arxiv","id":"1901.03300","version":3},"attestation_state":"computed","paper":{"title":"On Emergence and Complexity of Ergodic Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jairo Bochi, Pierre Berger","submitted_at":"2019-01-10T18:06:16Z","abstract_excerpt":"A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental propert"},"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":"1901.03300","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-10T18:06:16Z","cross_cats_sorted":[],"title_canon_sha256":"7c82eec6f4934b9126bab3343c3c231d5d6366834bef6f6e61c154c165fc2309","abstract_canon_sha256":"c94f3e08dcc2a5e8ab3e643b7370def32336c75c433d65965003e2f3a4991763"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:53:49.497228Z","signature_b64":"rwJzdvkWiMCD8oRImZEyf++SDerwEkSEyZlCsIh58XXBsUpxAVCAIHTlU8UpHEAeXa8KS1AMLrEhK4DHhjiWCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caf673eca5c1f8dda9d5c53f44f13adb3d686ce9d0f3604278bb392ac8264999","last_reissued_at":"2026-07-05T02:53:49.496703Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:53:49.496703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Emergence and Complexity of Ergodic Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jairo Bochi, Pierre Berger","submitted_at":"2019-01-10T18:06:16Z","abstract_excerpt":"A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental propert"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03300","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1901.03300/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1901.03300","created_at":"2026-07-05T02:53:49.496747+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.03300v3","created_at":"2026-07-05T02:53:49.496747+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.03300","created_at":"2026-07-05T02:53:49.496747+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZL3HH3FFYH4N","created_at":"2026-07-05T02:53:49.496747+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZL3HH3FFYH4N3KOV","created_at":"2026-07-05T02:53:49.496747+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZL3HH3FF","created_at":"2026-07-05T02:53:49.496747+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.00006","citing_title":"Ulam meets Turing: constructing quadratic maps with non-computable SRB measures","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M","json":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M.json","graph_json":"https://pith.science/api/pith-number/ZL3HH3FFYH4N3KOVYU7UJ4J23M/graph.json","events_json":"https://pith.science/api/pith-number/ZL3HH3FFYH4N3KOVYU7UJ4J23M/events.json","paper":"https://pith.science/paper/ZL3HH3FF"},"agent_actions":{"view_html":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M","download_json":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M.json","view_paper":"https://pith.science/paper/ZL3HH3FF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.03300&json=true","fetch_graph":"https://pith.science/api/pith-number/ZL3HH3FFYH4N3KOVYU7UJ4J23M/graph.json","fetch_events":"https://pith.science/api/pith-number/ZL3HH3FFYH4N3KOVYU7UJ4J23M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M/action/storage_attestation","attest_author":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M/action/author_attestation","sign_citation":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M/action/citation_signature","submit_replication":"https://pith.science/pith/ZL3HH3FFYH4N3KOVYU7UJ4J23M/action/replication_record"}},"created_at":"2026-07-05T02:53:49.496747+00:00","updated_at":"2026-07-05T02:53:49.496747+00:00"}