{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:O43RJ6ATP66HLBQGPKYXBAZVYC","short_pith_number":"pith:O43RJ6AT","schema_version":"1.0","canonical_sha256":"773714f8137fbc7586067ab1708335c0b33ef494a400cbbf3177b0b8c5b607d6","source":{"kind":"arxiv","id":"1507.00427","version":1},"attestation_state":"computed","paper":{"title":"On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein-Rutmann type Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yi Wang, Zeng Lian","submitted_at":"2015-07-02T05:07:48Z","abstract_excerpt":"For linear random dynamical systems in a separable Banach space $X$, we derived a series of Krein-Rutman type Theorems with respect to co-invariant cone family with rank-$k$, which present a (quasi)-equivalence relation between the measurably co-invariant cone family and the measurably dominated splitting of $X$. Moreover, such (quasi)-equivalence relation turns out to be an equivalence relation whenever (i) $k=1$; or (ii) in the frame of the Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater than the negative infinity.\n  For the second case, we thoroughly investigated"},"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":"1507.00427","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-02T05:07:48Z","cross_cats_sorted":[],"title_canon_sha256":"34db005c563d1939fd5ece628ed4e7aef7f515e2d18b83918748e82232d4c566","abstract_canon_sha256":"070d204c824eb9d666d332b2f64650e1073fe548c5106aa7299d33877e3ea01f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:24.783233Z","signature_b64":"LJET4dV5IV36EPSUFR/eYYKR1UuaUUfLNhHPWI9ZVAr4OAuiEIYyzDeRR9ieS3x08LgLfs0+dEgP3dhZB2lICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"773714f8137fbc7586067ab1708335c0b33ef494a400cbbf3177b0b8c5b607d6","last_reissued_at":"2026-05-18T01:37:24.782571Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:24.782571Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein-Rutmann type Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yi Wang, Zeng Lian","submitted_at":"2015-07-02T05:07:48Z","abstract_excerpt":"For linear random dynamical systems in a separable Banach space $X$, we derived a series of Krein-Rutman type Theorems with respect to co-invariant cone family with rank-$k$, which present a (quasi)-equivalence relation between the measurably co-invariant cone family and the measurably dominated splitting of $X$. Moreover, such (quasi)-equivalence relation turns out to be an equivalence relation whenever (i) $k=1$; or (ii) in the frame of the Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater than the negative infinity.\n  For the second case, we thoroughly investigated"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.00427","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":"1507.00427","created_at":"2026-05-18T01:37:24.782668+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.00427v1","created_at":"2026-05-18T01:37:24.782668+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.00427","created_at":"2026-05-18T01:37:24.782668+00:00"},{"alias_kind":"pith_short_12","alias_value":"O43RJ6ATP66H","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"O43RJ6ATP66HLBQG","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"O43RJ6AT","created_at":"2026-05-18T12:29:34.919912+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/O43RJ6ATP66HLBQGPKYXBAZVYC","json":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC.json","graph_json":"https://pith.science/api/pith-number/O43RJ6ATP66HLBQGPKYXBAZVYC/graph.json","events_json":"https://pith.science/api/pith-number/O43RJ6ATP66HLBQGPKYXBAZVYC/events.json","paper":"https://pith.science/paper/O43RJ6AT"},"agent_actions":{"view_html":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC","download_json":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC.json","view_paper":"https://pith.science/paper/O43RJ6AT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.00427&json=true","fetch_graph":"https://pith.science/api/pith-number/O43RJ6ATP66HLBQGPKYXBAZVYC/graph.json","fetch_events":"https://pith.science/api/pith-number/O43RJ6ATP66HLBQGPKYXBAZVYC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC/action/storage_attestation","attest_author":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC/action/author_attestation","sign_citation":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC/action/citation_signature","submit_replication":"https://pith.science/pith/O43RJ6ATP66HLBQGPKYXBAZVYC/action/replication_record"}},"created_at":"2026-05-18T01:37:24.782668+00:00","updated_at":"2026-05-18T01:37:24.782668+00:00"}