{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WQ347YFCO2Y7PVORUUJWLIQFEK","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":"816d65b93b3bbbe423d146262cd003b96e0994fab07aae3bdab8bc2e22d7f7cc","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T23:40:28Z","title_canon_sha256":"463880215ea790642370db86e13e31c1f665e8348f1bb8cb67ec52b5ca85a9cb"},"schema_version":"1.0","source":{"id":"1311.0927","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0927","created_at":"2026-05-18T02:47:31Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0927v4","created_at":"2026-05-18T02:47:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0927","created_at":"2026-05-18T02:47:31Z"},{"alias_kind":"pith_short_12","alias_value":"WQ347YFCO2Y7","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WQ347YFCO2Y7PVOR","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WQ347YFC","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:8dec106f01c373a2ba62fb19cfe17cd38b44620ba0f9f7eeac5f050df930cea7","target":"graph","created_at":"2026-05-18T02:47:31Z","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 consider two numerical entropy--type invariants for actions of $\\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. $\\Zk,\\,k\\ge 2$ actions on $k+1$-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of $\\mathbb Z$ actions the entropies of ergodic maximal","authors_text":"Anatole Katok, Federico Rodriguez Hertz, Svetlana Katok","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T23:40:28Z","title":"The Fried average entropy and slow entropy for actions of higher rank abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0927","kind":"arxiv","version":4},"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":"sha256:5b33147bf767209ffe52d702a9cfbbac86ea6457d82eaf6b558de8bde41ef620","target":"record","created_at":"2026-05-18T02:47:31Z","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":"816d65b93b3bbbe423d146262cd003b96e0994fab07aae3bdab8bc2e22d7f7cc","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T23:40:28Z","title_canon_sha256":"463880215ea790642370db86e13e31c1f665e8348f1bb8cb67ec52b5ca85a9cb"},"schema_version":"1.0","source":{"id":"1311.0927","kind":"arxiv","version":4}},"canonical_sha256":"b437cfe0a276b1f7d5d1a51365a20522aecb58c6d51566d89cb66dc91522d5b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b437cfe0a276b1f7d5d1a51365a20522aecb58c6d51566d89cb66dc91522d5b2","first_computed_at":"2026-05-18T02:47:31.445242Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:47:31.445242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nHKFuGfqCgjMkwbJsggxQ7oroyqQmwQDj5nAfBVKXSWg1FT1tZIBsslNjEbLxJNiuQOkqWNxPi7iZfrU/iD8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:47:31.445693Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.0927","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b33147bf767209ffe52d702a9cfbbac86ea6457d82eaf6b558de8bde41ef620","sha256:8dec106f01c373a2ba62fb19cfe17cd38b44620ba0f9f7eeac5f050df930cea7"],"state_sha256":"383fd98c9bdbd7a76721250160885e4337a036cc4fa100abafd692f70b09c1c9"}