{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:MMXAT2FBGPJ3FTRJPPD2OYLOXI","short_pith_number":"pith:MMXAT2FB","canonical_record":{"source":{"id":"1308.4019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-08-19T13:37:13Z","cross_cats_sorted":["math.DS","math.GR"],"title_canon_sha256":"63ec43d3b4f04bf343225305869e00672b6ab1d84b997967d832dfd1eaaf9aba","abstract_canon_sha256":"40ef89ef6b4964a40234812892034091a33e4855626363af3ace494094facd71"},"schema_version":"1.0"},"canonical_sha256":"632e09e8a133d3b2ce297bc7a7616eba04c8de64d6252544740b99429986e058","source":{"kind":"arxiv","id":"1308.4019","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.4019","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"arxiv_version","alias_value":"1308.4019v1","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4019","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"pith_short_12","alias_value":"MMXAT2FBGPJ3","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MMXAT2FBGPJ3FTRJ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MMXAT2FB","created_at":"2026-05-18T12:27:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:MMXAT2FBGPJ3FTRJPPD2OYLOXI","target":"record","payload":{"canonical_record":{"source":{"id":"1308.4019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-08-19T13:37:13Z","cross_cats_sorted":["math.DS","math.GR"],"title_canon_sha256":"63ec43d3b4f04bf343225305869e00672b6ab1d84b997967d832dfd1eaaf9aba","abstract_canon_sha256":"40ef89ef6b4964a40234812892034091a33e4855626363af3ace494094facd71"},"schema_version":"1.0"},"canonical_sha256":"632e09e8a133d3b2ce297bc7a7616eba04c8de64d6252544740b99429986e058","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:35.061507Z","signature_b64":"eV29gap36bHmwQkb/iSBFoP//1rJeaRM4i+yYJ/omyjhvUTHpHupLn0pIxN9dE8GC23wLpT3uSrSLm0c1jNvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"632e09e8a133d3b2ce297bc7a7616eba04c8de64d6252544740b99429986e058","last_reissued_at":"2026-05-18T03:15:35.060926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:35.060926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.4019","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:15:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pvr7jYwjTe+fgaRxyzM17o/rU8XNJr2DQsnGJv+EccnX46HT+4sw1p3XYcbs2e2ZAEMuUytGnyiQbP05+BuEBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:03:09.779752Z"},"content_sha256":"61fdf25a726ab343ebe2a7e4e5ab41dc18466239ca7764da93fe903f60e4312a","schema_version":"1.0","event_id":"sha256:61fdf25a726ab343ebe2a7e4e5ab41dc18466239ca7764da93fe903f60e4312a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:MMXAT2FBGPJ3FTRJPPD2OYLOXI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topological Entropy and Algebraic Entropy for group endomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR"],"primary_cat":"math.GN","authors_text":"Anna Giordano Bruno, Dikran Dikranjan","submitted_at":"2013-08-19T13:37:13Z","abstract_excerpt":"The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4019","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:15:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6GHnI1gjjz1tQ31DnzsTVxVdDFbyLhkWEwkfqXsG6ziHhMQWcLjXQAAYL5mAYOw3ASA7zc2gYNovd9CIjHarBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:03:09.780098Z"},"content_sha256":"d621ab0833130f42ef327f223e3f6a264efef04fff91411272c7d49a31d16e6d","schema_version":"1.0","event_id":"sha256:d621ab0833130f42ef327f223e3f6a264efef04fff91411272c7d49a31d16e6d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/bundle.json","state_url":"https://pith.science/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/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":"2026-06-01T12:03:09Z","links":{"resolver":"https://pith.science/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI","bundle":"https://pith.science/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/bundle.json","state":"https://pith.science/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MMXAT2FBGPJ3FTRJPPD2OYLOXI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:MMXAT2FBGPJ3FTRJPPD2OYLOXI","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":"40ef89ef6b4964a40234812892034091a33e4855626363af3ace494094facd71","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-08-19T13:37:13Z","title_canon_sha256":"63ec43d3b4f04bf343225305869e00672b6ab1d84b997967d832dfd1eaaf9aba"},"schema_version":"1.0","source":{"id":"1308.4019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.4019","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"arxiv_version","alias_value":"1308.4019v1","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4019","created_at":"2026-05-18T03:15:35Z"},{"alias_kind":"pith_short_12","alias_value":"MMXAT2FBGPJ3","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MMXAT2FBGPJ3FTRJ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MMXAT2FB","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:d621ab0833130f42ef327f223e3f6a264efef04fff91411272c7d49a31d16e6d","target":"graph","created_at":"2026-05-18T03:15:35Z","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":"The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Math","authors_text":"Anna Giordano Bruno, Dikran Dikranjan","cross_cats":["math.DS","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-08-19T13:37:13Z","title":"Topological Entropy and Algebraic Entropy for group endomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4019","kind":"arxiv","version":1},"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":"sha256:61fdf25a726ab343ebe2a7e4e5ab41dc18466239ca7764da93fe903f60e4312a","target":"record","created_at":"2026-05-18T03:15:35Z","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":"40ef89ef6b4964a40234812892034091a33e4855626363af3ace494094facd71","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2013-08-19T13:37:13Z","title_canon_sha256":"63ec43d3b4f04bf343225305869e00672b6ab1d84b997967d832dfd1eaaf9aba"},"schema_version":"1.0","source":{"id":"1308.4019","kind":"arxiv","version":1}},"canonical_sha256":"632e09e8a133d3b2ce297bc7a7616eba04c8de64d6252544740b99429986e058","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"632e09e8a133d3b2ce297bc7a7616eba04c8de64d6252544740b99429986e058","first_computed_at":"2026-05-18T03:15:35.060926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:35.060926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eV29gap36bHmwQkb/iSBFoP//1rJeaRM4i+yYJ/omyjhvUTHpHupLn0pIxN9dE8GC23wLpT3uSrSLm0c1jNvCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:35.061507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.4019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61fdf25a726ab343ebe2a7e4e5ab41dc18466239ca7764da93fe903f60e4312a","sha256:d621ab0833130f42ef327f223e3f6a264efef04fff91411272c7d49a31d16e6d"],"state_sha256":"072b2fe0a19fa0cb41e09b3951510d0bd7d46bd460706a2323f9dc1204289de1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mdu+6j5LBmCojeph+JXwUvbqO3a2nNeLx9WMLrXpAC0A9y2pb+SXH0VIvkEf+7k5APdMqiVhPdeJ0Rv1trO2DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T12:03:09.782173Z","bundle_sha256":"6f943e160c317cf90a115ce120877faee24011d60bbd4c247afbf10168c48a61"}}