{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6FLUHOXOBOPNDK56VZR7RGZOEA","short_pith_number":"pith:6FLUHOXO","canonical_record":{"source":{"id":"1105.4344","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-22T15:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"ff9c648440a57e17e743c39c00d459673e25ca0d2fea04adff645ec88634babd","abstract_canon_sha256":"85c47c5d6e7b3a37f34052e10da1332f9c5254a2d75febf68f5997a4ae003f96"},"schema_version":"1.0"},"canonical_sha256":"f15743baee0b9ed1abbeae63f89b2e2039775436e8087590b0ce570aa58f832a","source":{"kind":"arxiv","id":"1105.4344","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.4344","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"arxiv_version","alias_value":"1105.4344v2","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4344","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"pith_short_12","alias_value":"6FLUHOXOBOPN","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6FLUHOXOBOPNDK56","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6FLUHOXO","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6FLUHOXOBOPNDK56VZR7RGZOEA","target":"record","payload":{"canonical_record":{"source":{"id":"1105.4344","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-22T15:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"ff9c648440a57e17e743c39c00d459673e25ca0d2fea04adff645ec88634babd","abstract_canon_sha256":"85c47c5d6e7b3a37f34052e10da1332f9c5254a2d75febf68f5997a4ae003f96"},"schema_version":"1.0"},"canonical_sha256":"f15743baee0b9ed1abbeae63f89b2e2039775436e8087590b0ce570aa58f832a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:45.173618Z","signature_b64":"f8JfqmbgNc3IcNQBbj33dmgsJ4hy6rJVKgVh2Ygxq2AUkjthEDN/o09iu631g/F5ogg/I+vLEW19so/7ew+6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f15743baee0b9ed1abbeae63f89b2e2039775436e8087590b0ce570aa58f832a","last_reissued_at":"2026-05-18T03:48:45.172791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:45.172791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.4344","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:48:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YQv+i3IUtRCMhqawjbppdRXYB3zAlkI2dmPSLlB0lIrtDvaFiA5VPMTdY3fK73MOz5HaxxcZ1lTDXqa1n2iaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:13:47.055409Z"},"content_sha256":"85c9ea206f8135a333b579daefcd638ad7d42dca7fdf47e043378b4afec49a9a","schema_version":"1.0","event_id":"sha256:85c9ea206f8135a333b579daefcd638ad7d42dca7fdf47e043378b4afec49a9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6FLUHOXOBOPNDK56VZR7RGZOEA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy of Endomorphisms of Lie Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Andr\\'e Caldas, Mauro Patr\\~ao","submitted_at":"2011-05-22T15:55:34Z","abstract_excerpt":"We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4344","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:48:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RXKN/RJlnuOiiIqG8JkilY+ceMHbrQMC7H6O26ci87t79smWfjVAirhfnW3jwaVNi5/q7OeQ0P49dFPkBUyIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:13:47.055749Z"},"content_sha256":"21f5d7474d46558c1b0f73bb3b9638e4673d27e1475d4c27fea2a5005dffc391","schema_version":"1.0","event_id":"sha256:21f5d7474d46558c1b0f73bb3b9638e4673d27e1475d4c27fea2a5005dffc391"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/bundle.json","state_url":"https://pith.science/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T12:13:47Z","links":{"resolver":"https://pith.science/pith/6FLUHOXOBOPNDK56VZR7RGZOEA","bundle":"https://pith.science/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/bundle.json","state":"https://pith.science/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6FLUHOXOBOPNDK56VZR7RGZOEA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6FLUHOXOBOPNDK56VZR7RGZOEA","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":"85c47c5d6e7b3a37f34052e10da1332f9c5254a2d75febf68f5997a4ae003f96","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-22T15:55:34Z","title_canon_sha256":"ff9c648440a57e17e743c39c00d459673e25ca0d2fea04adff645ec88634babd"},"schema_version":"1.0","source":{"id":"1105.4344","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.4344","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"arxiv_version","alias_value":"1105.4344v2","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4344","created_at":"2026-05-18T03:48:45Z"},{"alias_kind":"pith_short_12","alias_value":"6FLUHOXOBOPN","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6FLUHOXOBOPNDK56","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6FLUHOXO","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:21f5d7474d46558c1b0f73bb3b9638e4673d27e1475d4c27fea2a5005dffc391","target":"graph","created_at":"2026-05-18T03:48:45Z","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 show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus.","authors_text":"Andr\\'e Caldas, Mauro Patr\\~ao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-22T15:55:34Z","title":"Entropy of Endomorphisms of Lie Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4344","kind":"arxiv","version":2},"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":"sha256:85c9ea206f8135a333b579daefcd638ad7d42dca7fdf47e043378b4afec49a9a","target":"record","created_at":"2026-05-18T03:48:45Z","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":"85c47c5d6e7b3a37f34052e10da1332f9c5254a2d75febf68f5997a4ae003f96","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-05-22T15:55:34Z","title_canon_sha256":"ff9c648440a57e17e743c39c00d459673e25ca0d2fea04adff645ec88634babd"},"schema_version":"1.0","source":{"id":"1105.4344","kind":"arxiv","version":2}},"canonical_sha256":"f15743baee0b9ed1abbeae63f89b2e2039775436e8087590b0ce570aa58f832a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f15743baee0b9ed1abbeae63f89b2e2039775436e8087590b0ce570aa58f832a","first_computed_at":"2026-05-18T03:48:45.172791Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:45.172791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f8JfqmbgNc3IcNQBbj33dmgsJ4hy6rJVKgVh2Ygxq2AUkjthEDN/o09iu631g/F5ogg/I+vLEW19so/7ew+6Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:45.173618Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.4344","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:85c9ea206f8135a333b579daefcd638ad7d42dca7fdf47e043378b4afec49a9a","sha256:21f5d7474d46558c1b0f73bb3b9638e4673d27e1475d4c27fea2a5005dffc391"],"state_sha256":"a3a0c7c4ec9c368e1fb0b0663f5982fd01753c7e064b57e6512e4dac179d695d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OcOxjbahxFNWwIBBXMGK28t3yEi8cj/PoDxioZJhlOX7LLZtHoBzXFH1rOa3cNkmTzM2XVbWYZFh4WYuZAuVDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T12:13:47.057570Z","bundle_sha256":"42b4b3c0642b3def82dfdb49110209b67b4e7146f4c61429470fe5e4ce49cae0"}}