{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:AB5VJGU2DASZ7RSC34B3O3DTD6","short_pith_number":"pith:AB5VJGU2","schema_version":"1.0","canonical_sha256":"007b549a9a18259fc642df03b76c731fa8addc6082d025a70aa0ea0822779326","source":{"kind":"arxiv","id":"1706.05453","version":1},"attestation_state":"computed","paper":{"title":"A note on two types of Lyapunov exponents and entropies for $\\mathbb{Z}^k$-actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yujun Zhu","submitted_at":"2017-06-16T23:27:57Z","abstract_excerpt":"In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $\\mathbb{Z}^k$-actions. The close relation among these quantities are investigated and the formulas of them are given via the Lyapunov exponents of the generators."},"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":"1706.05453","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-06-16T23:27:57Z","cross_cats_sorted":[],"title_canon_sha256":"c705162222100770f7ce191e798e8dac541a0423ed77d0a28c8a7f5ac87d73d7","abstract_canon_sha256":"c0a5c4820df451807fa8dab1a12a7aa7c2064cecb5a18c805014336ba71db3e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:11.585431Z","signature_b64":"Np0D42mbuaTR6B+Tk8W9UGulljhZtNG7qEUbrIFZsz8UBB7+hqYJQOLcTvHmZAHVTV2Vl1U6gFpvGBPCxcXnAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"007b549a9a18259fc642df03b76c731fa8addc6082d025a70aa0ea0822779326","last_reissued_at":"2026-05-18T00:42:11.584919Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:11.584919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on two types of Lyapunov exponents and entropies for $\\mathbb{Z}^k$-actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yujun Zhu","submitted_at":"2017-06-16T23:27:57Z","abstract_excerpt":"In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $\\mathbb{Z}^k$-actions. The close relation among these quantities are investigated and the formulas of them are given via the Lyapunov exponents of the generators."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05453","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":"1706.05453","created_at":"2026-05-18T00:42:11.585009+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.05453v1","created_at":"2026-05-18T00:42:11.585009+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05453","created_at":"2026-05-18T00:42:11.585009+00:00"},{"alias_kind":"pith_short_12","alias_value":"AB5VJGU2DASZ","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AB5VJGU2DASZ7RSC","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AB5VJGU2","created_at":"2026-05-18T12:31:05.417338+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/AB5VJGU2DASZ7RSC34B3O3DTD6","json":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6.json","graph_json":"https://pith.science/api/pith-number/AB5VJGU2DASZ7RSC34B3O3DTD6/graph.json","events_json":"https://pith.science/api/pith-number/AB5VJGU2DASZ7RSC34B3O3DTD6/events.json","paper":"https://pith.science/paper/AB5VJGU2"},"agent_actions":{"view_html":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6","download_json":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6.json","view_paper":"https://pith.science/paper/AB5VJGU2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.05453&json=true","fetch_graph":"https://pith.science/api/pith-number/AB5VJGU2DASZ7RSC34B3O3DTD6/graph.json","fetch_events":"https://pith.science/api/pith-number/AB5VJGU2DASZ7RSC34B3O3DTD6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6/action/storage_attestation","attest_author":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6/action/author_attestation","sign_citation":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6/action/citation_signature","submit_replication":"https://pith.science/pith/AB5VJGU2DASZ7RSC34B3O3DTD6/action/replication_record"}},"created_at":"2026-05-18T00:42:11.585009+00:00","updated_at":"2026-05-18T00:42:11.585009+00:00"}