{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:LRXY4UWD4BGPQKIHSZ25NHPWK2","short_pith_number":"pith:LRXY4UWD","schema_version":"1.0","canonical_sha256":"5c6f8e52c3e04cf829079675d69df656bd4c426bce276694970ae14042e3e8b1","source":{"kind":"arxiv","id":"1504.04705","version":1},"attestation_state":"computed","paper":{"title":"On Bertelson-Gromov Dynamical Morse Entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.AT","math.MP"],"primary_cat":"math.DS","authors_text":"Artur O. Lopes, Marcos Sebastiani","submitted_at":"2015-04-18T10:24:56Z","abstract_excerpt":"In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\\infty$ manifold of finite dimension. Assume that $f_0 :M \\to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \\in M^n$, denote $ f_n (x) = \\frac{1}{n} \\, \\sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\\subset [0,1]"},"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":"1504.04705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-18T10:24:56Z","cross_cats_sorted":["cond-mat.stat-mech","math-ph","math.AT","math.MP"],"title_canon_sha256":"5087107b3d3467acd7e618fb64a3cdf86709dfa7ce5abd52414d6248e7bf1b59","abstract_canon_sha256":"033416976385f0c60a9b9454aa6cc7e3e642266efe77a215fce947e56295712e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:25.060763Z","signature_b64":"nT72p7k38XyuZkcfCM78+ohAdcNzFPyoCO4iR2pyziH31sdDg6y/r1AwQ71wbyneN//02xUOUhzj9l2KroDPAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c6f8e52c3e04cf829079675d69df656bd4c426bce276694970ae14042e3e8b1","last_reissued_at":"2026-05-18T02:18:25.060268Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:25.060268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Bertelson-Gromov Dynamical Morse Entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.AT","math.MP"],"primary_cat":"math.DS","authors_text":"Artur O. Lopes, Marcos Sebastiani","submitted_at":"2015-04-18T10:24:56Z","abstract_excerpt":"In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\\infty$ manifold of finite dimension. Assume that $f_0 :M \\to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \\in M^n$, denote $ f_n (x) = \\frac{1}{n} \\, \\sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\\subset [0,1]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04705","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":"1504.04705","created_at":"2026-05-18T02:18:25.060351+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.04705v1","created_at":"2026-05-18T02:18:25.060351+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04705","created_at":"2026-05-18T02:18:25.060351+00:00"},{"alias_kind":"pith_short_12","alias_value":"LRXY4UWD4BGP","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"LRXY4UWD4BGPQKIH","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"LRXY4UWD","created_at":"2026-05-18T12:29:29.992203+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/LRXY4UWD4BGPQKIHSZ25NHPWK2","json":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2.json","graph_json":"https://pith.science/api/pith-number/LRXY4UWD4BGPQKIHSZ25NHPWK2/graph.json","events_json":"https://pith.science/api/pith-number/LRXY4UWD4BGPQKIHSZ25NHPWK2/events.json","paper":"https://pith.science/paper/LRXY4UWD"},"agent_actions":{"view_html":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2","download_json":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2.json","view_paper":"https://pith.science/paper/LRXY4UWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.04705&json=true","fetch_graph":"https://pith.science/api/pith-number/LRXY4UWD4BGPQKIHSZ25NHPWK2/graph.json","fetch_events":"https://pith.science/api/pith-number/LRXY4UWD4BGPQKIHSZ25NHPWK2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2/action/storage_attestation","attest_author":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2/action/author_attestation","sign_citation":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2/action/citation_signature","submit_replication":"https://pith.science/pith/LRXY4UWD4BGPQKIHSZ25NHPWK2/action/replication_record"}},"created_at":"2026-05-18T02:18:25.060351+00:00","updated_at":"2026-05-18T02:18:25.060351+00:00"}