{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7YWDJW3NI4ERQWYRDM26C3BZDJ","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":"d4009751eb2975acb458d643495a8529a9522c6d656d9a4d9bb1f1f84a532938","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-02-09T08:28:38Z","title_canon_sha256":"4b11924bfda6f3610096fea27127f2760831444449e508eb091f4040a823c044"},"schema_version":"1.0","source":{"id":"1602.02894","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.02894","created_at":"2026-05-18T01:21:06Z"},{"alias_kind":"arxiv_version","alias_value":"1602.02894v1","created_at":"2026-05-18T01:21:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.02894","created_at":"2026-05-18T01:21:06Z"},{"alias_kind":"pith_short_12","alias_value":"7YWDJW3NI4ER","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7YWDJW3NI4ERQWYR","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7YWDJW3N","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:81de35a65892d1ae77563cfcd67c47dc766ebae113a20d7b5e80435ce54f914e","target":"graph","created_at":"2026-05-18T01:21:06Z","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 discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.","authors_text":"Nadav Dym","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-02-09T08:28:38Z","title":"Spatial Recurrence for Ergodic Fractal Measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02894","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":"sha256:95b71b17671926ae9cd40d12064f65c3a7f7569f45ddba428d944fe580b811d4","target":"record","created_at":"2026-05-18T01:21:06Z","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":"d4009751eb2975acb458d643495a8529a9522c6d656d9a4d9bb1f1f84a532938","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-02-09T08:28:38Z","title_canon_sha256":"4b11924bfda6f3610096fea27127f2760831444449e508eb091f4040a823c044"},"schema_version":"1.0","source":{"id":"1602.02894","kind":"arxiv","version":1}},"canonical_sha256":"fe2c34db6d4709185b111b35e16c391a79a4ec8db5b952552e6ffa86ab134df9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe2c34db6d4709185b111b35e16c391a79a4ec8db5b952552e6ffa86ab134df9","first_computed_at":"2026-05-18T01:21:06.164480Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:06.164480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UmcnX7IQOLun++7RI8RsI401cm9F0qSuGsHz1IFXuJMuzRM6Ss1mN2MYmuTUz7HnPmmyOCwvMtRYJU2xePQyDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:06.165087Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.02894","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95b71b17671926ae9cd40d12064f65c3a7f7569f45ddba428d944fe580b811d4","sha256:81de35a65892d1ae77563cfcd67c47dc766ebae113a20d7b5e80435ce54f914e"],"state_sha256":"ad26a244cfe3bd8bb540e9a0fcd903faf7be3fc553f2db94eae37457a29f5ad4"}