{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:O3ZBODL2QC3TE6XFPFNUGLT3FE","short_pith_number":"pith:O3ZBODL2","canonical_record":{"source":{"id":"1711.02562","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2a243c7546e6b5a244cddd9c64712b3b1bc0436f04a142c45112abc7ca39b220","abstract_canon_sha256":"81277f2cf0713fdb393910670c0a6daeff332ec4b22a4d857a82f4f7a2a0232e"},"schema_version":"1.0"},"canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","source":{"kind":"arxiv","id":"1711.02562","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.02562","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"arxiv_version","alias_value":"1711.02562v4","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.02562","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"pith_short_12","alias_value":"O3ZBODL2QC3T","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"O3ZBODL2QC3TE6XF","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"O3ZBODL2","created_at":"2026-05-18T12:31:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:O3ZBODL2QC3TE6XFPFNUGLT3FE","target":"record","payload":{"canonical_record":{"source":{"id":"1711.02562","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2a243c7546e6b5a244cddd9c64712b3b1bc0436f04a142c45112abc7ca39b220","abstract_canon_sha256":"81277f2cf0713fdb393910670c0a6daeff332ec4b22a4d857a82f4f7a2a0232e"},"schema_version":"1.0"},"canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:18.958848Z","signature_b64":"4lgBq9tTU7+BWdTjmEkYGXzr8Im0f0AlQTY7Kueza++XUigw2Vywk2HeAiRUr8CSO8M+eTGB+iJZAcLrbafZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","last_reissued_at":"2026-05-18T00:17:18.958406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:18.958406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.02562","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fst3qeEYxOelF7Re8GdT5NKp1eKWoxL2N7XL3gckfVKc3amfAj/Eq6N23xeg6050Ps/KLwtjsrcUYRgFInTLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:19:07.517267Z"},"content_sha256":"2cb32bef4ef97b9a8556a0fd7f53eba0070023ac1e0a1714ec6b13c8525b191a","schema_version":"1.0","event_id":"sha256:2cb32bef4ef97b9a8556a0fd7f53eba0070023ac1e0a1714ec6b13c8525b191a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:O3ZBODL2QC3TE6XFPFNUGLT3FE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The topological entropy of endomorphisms of Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Mauro Patr\\~ao","submitted_at":"2017-11-06T18:44:37Z","abstract_excerpt":"In this paper, we determine the topological entropy $h(\\phi)$ of a continuous endomorphism $\\phi$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\\phi)$, which is characterized by the so called variational principle. We prove that \\[ h\\left(\\phi\\right) = h\\left(\\phi|_{T(G_\\phi)}\\right) \\] where $G_\\phi$ is the maximal connected subgroup of $G$ such that $\\phi(G_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02562","kind":"arxiv","version":4},"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":"2026-05-18T00:17:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bdgh+W3o8NXgRJRBTqE6AyA/u8713att/ySk4FDkZJx0ZpxttsDRc9J4k7Ex98EbDYLJVzV4sOI7ry8KrYMsAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:19:07.517638Z"},"content_sha256":"2f335f2f2738553998e35fdc7c68d5175c9b23411a8fed044758a429900c1959","schema_version":"1.0","event_id":"sha256:2f335f2f2738553998e35fdc7c68d5175c9b23411a8fed044758a429900c1959"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/bundle.json","state_url":"https://pith.science/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/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":"2026-06-22T03:19:07Z","links":{"resolver":"https://pith.science/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE","bundle":"https://pith.science/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/bundle.json","state":"https://pith.science/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O3ZBODL2QC3TE6XFPFNUGLT3FE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:O3ZBODL2QC3TE6XFPFNUGLT3FE","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":"81277f2cf0713fdb393910670c0a6daeff332ec4b22a4d857a82f4f7a2a0232e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","title_canon_sha256":"2a243c7546e6b5a244cddd9c64712b3b1bc0436f04a142c45112abc7ca39b220"},"schema_version":"1.0","source":{"id":"1711.02562","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.02562","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"arxiv_version","alias_value":"1711.02562v4","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.02562","created_at":"2026-05-18T00:17:18Z"},{"alias_kind":"pith_short_12","alias_value":"O3ZBODL2QC3T","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"O3ZBODL2QC3TE6XF","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"O3ZBODL2","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:2f335f2f2738553998e35fdc7c68d5175c9b23411a8fed044758a429900c1959","target":"graph","created_at":"2026-05-18T00:17:18Z","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":"In this paper, we determine the topological entropy $h(\\phi)$ of a continuous endomorphism $\\phi$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\\phi)$, which is characterized by the so called variational principle. We prove that \\[ h\\left(\\phi\\right) = h\\left(\\phi|_{T(G_\\phi)}\\right) \\] where $G_\\phi$ is the maximal connected subgroup of $G$ such that $\\phi(G_","authors_text":"Mauro Patr\\~ao","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","title":"The topological entropy of endomorphisms of Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02562","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":"sha256:2cb32bef4ef97b9a8556a0fd7f53eba0070023ac1e0a1714ec6b13c8525b191a","target":"record","created_at":"2026-05-18T00:17:18Z","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":"81277f2cf0713fdb393910670c0a6daeff332ec4b22a4d857a82f4f7a2a0232e","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","title_canon_sha256":"2a243c7546e6b5a244cddd9c64712b3b1bc0436f04a142c45112abc7ca39b220"},"schema_version":"1.0","source":{"id":"1711.02562","kind":"arxiv","version":4}},"canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","first_computed_at":"2026-05-18T00:17:18.958406Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:18.958406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4lgBq9tTU7+BWdTjmEkYGXzr8Im0f0AlQTY7Kueza++XUigw2Vywk2HeAiRUr8CSO8M+eTGB+iJZAcLrbafZBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:18.958848Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.02562","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2cb32bef4ef97b9a8556a0fd7f53eba0070023ac1e0a1714ec6b13c8525b191a","sha256:2f335f2f2738553998e35fdc7c68d5175c9b23411a8fed044758a429900c1959"],"state_sha256":"fb9b01d2920d48b138593dfa0c85c3fb34edf71fff63b75dfcb16c1e7bee940d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BYsr2lp6JpuLHZjr6y5U5haiEvqGZv4Y3nKEep+m1cKzHI597HvA3k5ssBz8kSHGyEVlUh72xxc7MG9IWLiOBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T03:19:07.519543Z","bundle_sha256":"307d952decbffa9a9f4678f220c7f852aa1f106a144443f1dd85c068bb3a0d27"}}