{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:EB4NSZRYCNHV7T6MVMI7LG2GH3","short_pith_number":"pith:EB4NSZRY","canonical_record":{"source":{"id":"1208.4550","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-22T16:33:52Z","cross_cats_sorted":[],"title_canon_sha256":"2d6170f9a176be9d681199595ccb831a18c68fb92d414fcb7df6486e4463408c","abstract_canon_sha256":"e99aed918bd2cae8f75d89b4edfb4c5cac472fc1e1c4b4f6d6c89645e4844bc5"},"schema_version":"1.0"},"canonical_sha256":"2078d96638134f5fcfccab11f59b463ee94039df67dd3bf3a8f584e2528db98f","source":{"kind":"arxiv","id":"1208.4550","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4550","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4550v1","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4550","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"EB4NSZRYCNHV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EB4NSZRYCNHV7T6M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EB4NSZRY","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:EB4NSZRYCNHV7T6MVMI7LG2GH3","target":"record","payload":{"canonical_record":{"source":{"id":"1208.4550","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-22T16:33:52Z","cross_cats_sorted":[],"title_canon_sha256":"2d6170f9a176be9d681199595ccb831a18c68fb92d414fcb7df6486e4463408c","abstract_canon_sha256":"e99aed918bd2cae8f75d89b4edfb4c5cac472fc1e1c4b4f6d6c89645e4844bc5"},"schema_version":"1.0"},"canonical_sha256":"2078d96638134f5fcfccab11f59b463ee94039df67dd3bf3a8f584e2528db98f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:00.716392Z","signature_b64":"Qn0mhxCEJQRqZGsmGC6FRTSXeMyayYWM1mqtp9n2a60JBBdfkyRVq/MsqCbYdqDefGmV7v7NNcJe+jdsC00VDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2078d96638134f5fcfccab11f59b463ee94039df67dd3bf3a8f584e2528db98f","last_reissued_at":"2026-05-18T02:21:00.715793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:00.715793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.4550","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":"2026-05-18T02:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"quLqFD9iSe60J8GnzxxIPJCEWQIi4gn6yPeicr0wtOtNHmte9yUtLnQHiiDxxsL6NKBChlvOj3we2tm7hsfvCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:06:23.377141Z"},"content_sha256":"2b4143aba94ceb02d3452ff212a711e6216c2dfad8c390e324e8574f7de3a6c5","schema_version":"1.0","event_id":"sha256:2b4143aba94ceb02d3452ff212a711e6216c2dfad8c390e324e8574f7de3a6c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:EB4NSZRYCNHV7T6MVMI7LG2GH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Measure theory through dynamical eyes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anatole Katok, Vaughn Climenhaga","submitted_at":"2012-08-22T16:33:52Z","abstract_excerpt":"These expository notes are a somewhat embellished version of two rather informal evening review sessions given by the second author at the 2008 Bedlewo summer school on \"Dynamical Systems - Geometric Structures and Rigidity\"; they provide a brief overview of some of the basics of measure theory and its applications to dynamics which are foundational to the various courses at the school."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4550","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":"2026-05-18T02:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mEehH+92Efr57wLs5nB7XDnoSoo4swkpWq1H96gx/khaWfrex5mKZJk0jaeyRQEuGlMdVyP7//P+wW2CTrEDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:06:23.377497Z"},"content_sha256":"63aa054bc7f98536529328c27b995516441f7bd33885b9fff93a4d86f18cb05e","schema_version":"1.0","event_id":"sha256:63aa054bc7f98536529328c27b995516441f7bd33885b9fff93a4d86f18cb05e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/bundle.json","state_url":"https://pith.science/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/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":"2026-06-02T12:06:23Z","links":{"resolver":"https://pith.science/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3","bundle":"https://pith.science/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/bundle.json","state":"https://pith.science/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EB4NSZRYCNHV7T6MVMI7LG2GH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:EB4NSZRYCNHV7T6MVMI7LG2GH3","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":"e99aed918bd2cae8f75d89b4edfb4c5cac472fc1e1c4b4f6d6c89645e4844bc5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-22T16:33:52Z","title_canon_sha256":"2d6170f9a176be9d681199595ccb831a18c68fb92d414fcb7df6486e4463408c"},"schema_version":"1.0","source":{"id":"1208.4550","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4550","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4550v1","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4550","created_at":"2026-05-18T02:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"EB4NSZRYCNHV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EB4NSZRYCNHV7T6M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EB4NSZRY","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:63aa054bc7f98536529328c27b995516441f7bd33885b9fff93a4d86f18cb05e","target":"graph","created_at":"2026-05-18T02:21:00Z","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":"These expository notes are a somewhat embellished version of two rather informal evening review sessions given by the second author at the 2008 Bedlewo summer school on \"Dynamical Systems - Geometric Structures and Rigidity\"; they provide a brief overview of some of the basics of measure theory and its applications to dynamics which are foundational to the various courses at the school.","authors_text":"Anatole Katok, Vaughn Climenhaga","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-22T16:33:52Z","title":"Measure theory through dynamical eyes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4550","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":"sha256:2b4143aba94ceb02d3452ff212a711e6216c2dfad8c390e324e8574f7de3a6c5","target":"record","created_at":"2026-05-18T02:21:00Z","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":"e99aed918bd2cae8f75d89b4edfb4c5cac472fc1e1c4b4f6d6c89645e4844bc5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-22T16:33:52Z","title_canon_sha256":"2d6170f9a176be9d681199595ccb831a18c68fb92d414fcb7df6486e4463408c"},"schema_version":"1.0","source":{"id":"1208.4550","kind":"arxiv","version":1}},"canonical_sha256":"2078d96638134f5fcfccab11f59b463ee94039df67dd3bf3a8f584e2528db98f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2078d96638134f5fcfccab11f59b463ee94039df67dd3bf3a8f584e2528db98f","first_computed_at":"2026-05-18T02:21:00.715793Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:00.715793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qn0mhxCEJQRqZGsmGC6FRTSXeMyayYWM1mqtp9n2a60JBBdfkyRVq/MsqCbYdqDefGmV7v7NNcJe+jdsC00VDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:00.716392Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4550","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b4143aba94ceb02d3452ff212a711e6216c2dfad8c390e324e8574f7de3a6c5","sha256:63aa054bc7f98536529328c27b995516441f7bd33885b9fff93a4d86f18cb05e"],"state_sha256":"5fc6b6d579059a6bb8ce10a88962aa4f5f6aa1298a08c811a175184262fa08ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d/E1sQggqqCy/3cUPrsgLdR66tvX4Dh4wn4a2l79pOGpx5J4aMquw5eNfda/Ed7B/B/W1G+bxqc3vgbcCLkGDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:06:23.379507Z","bundle_sha256":"2ad70c58fe545db5545a43c58883c2985e7f5d589a94cfb5e25243f28f7d6b8c"}}