{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PZHNEF4KSN3AXP24JKBZDK3GHJ","short_pith_number":"pith:PZHNEF4K","canonical_record":{"source":{"id":"1502.02693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T21:22:44Z","cross_cats_sorted":[],"title_canon_sha256":"781d73f9f88aec6314f06215d1c3f28eabcbf5781fee067ad0a4ba2e20b8932c","abstract_canon_sha256":"c3f006a9b05ad5f46af417ec5a8b28e88dd2ac95c056dfc788edd483c1dd5c2e"},"schema_version":"1.0"},"canonical_sha256":"7e4ed2178a93760bbf5c4a8391ab663a42876b80a70c2f5a470ea1da7f1ff12b","source":{"kind":"arxiv","id":"1502.02693","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.02693","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"1502.02693v2","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.02693","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"PZHNEF4KSN3A","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PZHNEF4KSN3AXP24","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PZHNEF4K","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PZHNEF4KSN3AXP24JKBZDK3GHJ","target":"record","payload":{"canonical_record":{"source":{"id":"1502.02693","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T21:22:44Z","cross_cats_sorted":[],"title_canon_sha256":"781d73f9f88aec6314f06215d1c3f28eabcbf5781fee067ad0a4ba2e20b8932c","abstract_canon_sha256":"c3f006a9b05ad5f46af417ec5a8b28e88dd2ac95c056dfc788edd483c1dd5c2e"},"schema_version":"1.0"},"canonical_sha256":"7e4ed2178a93760bbf5c4a8391ab663a42876b80a70c2f5a470ea1da7f1ff12b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:45.552643Z","signature_b64":"/gvenE1kxw3QQtembdwD2TWgWNo1P6piLoISilDJnStCvoMKHrww44+/8dVfnbB2+BfFFGx8J6pzYE4ynEW0BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e4ed2178a93760bbf5c4a8391ab663a42876b80a70c2f5a470ea1da7f1ff12b","last_reissued_at":"2026-05-18T01:14:45.551909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:45.551909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.02693","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":"2026-05-18T01:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IyFat0OcqvoiDUpOfJfNlXO7iO4LWB3hfKoTs7vPf366Yp+qqjj+T52ejDRf9kjqSquMFJRRsW9XjkX2IWzfBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:37:10.261109Z"},"content_sha256":"66a8f3ee6d9a315226b96a23287a44afbf6b392a94eb1c110eca10c3d5de7f85","schema_version":"1.0","event_id":"sha256:66a8f3ee6d9a315226b96a23287a44afbf6b392a94eb1c110eca10c3d5de7f85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PZHNEF4KSN3AXP24JKBZDK3GHJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Closed orbits in quotient systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Stefanie Zegowitz","submitted_at":"2015-02-09T21:22:44Z","abstract_excerpt":"We study the relationship between pairs of topological dynamical systems $ (X,T) $ and $ (X',T') $, where $ (X',T') $ is the quotient of $ (X,T) $ under the action of a finite group $ G $. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of $ G $ and show that any growth rate in between these bounds can be achieved."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02693","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":"2026-05-18T01:14:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BvGjlcvpN0bo1x01MBoVFPGhRn+JOgFGqvsenibSfMbVOdZhyGxaL8EmQYmb9jgzGfWBRXmCaKes91F3AWMLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:37:10.261840Z"},"content_sha256":"a2a3ecfa1215d30193af7b5ac0e02b15d4ed9598329122e6b2e891e4b368ad19","schema_version":"1.0","event_id":"sha256:a2a3ecfa1215d30193af7b5ac0e02b15d4ed9598329122e6b2e891e4b368ad19"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/bundle.json","state_url":"https://pith.science/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/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":"2026-06-09T17:37:10Z","links":{"resolver":"https://pith.science/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ","bundle":"https://pith.science/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/bundle.json","state":"https://pith.science/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PZHNEF4KSN3AXP24JKBZDK3GHJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PZHNEF4KSN3AXP24JKBZDK3GHJ","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":"c3f006a9b05ad5f46af417ec5a8b28e88dd2ac95c056dfc788edd483c1dd5c2e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T21:22:44Z","title_canon_sha256":"781d73f9f88aec6314f06215d1c3f28eabcbf5781fee067ad0a4ba2e20b8932c"},"schema_version":"1.0","source":{"id":"1502.02693","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.02693","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"arxiv_version","alias_value":"1502.02693v2","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.02693","created_at":"2026-05-18T01:14:45Z"},{"alias_kind":"pith_short_12","alias_value":"PZHNEF4KSN3A","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PZHNEF4KSN3AXP24","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PZHNEF4K","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:a2a3ecfa1215d30193af7b5ac0e02b15d4ed9598329122e6b2e891e4b368ad19","target":"graph","created_at":"2026-05-18T01:14:45Z","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 study the relationship between pairs of topological dynamical systems $ (X,T) $ and $ (X',T') $, where $ (X',T') $ is the quotient of $ (X,T) $ under the action of a finite group $ G $. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of $ G $ and show that any growth rate in between these bounds can be achieved.","authors_text":"Stefanie Zegowitz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T21:22:44Z","title":"Closed orbits in quotient systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.02693","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":"sha256:66a8f3ee6d9a315226b96a23287a44afbf6b392a94eb1c110eca10c3d5de7f85","target":"record","created_at":"2026-05-18T01:14:45Z","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":"c3f006a9b05ad5f46af417ec5a8b28e88dd2ac95c056dfc788edd483c1dd5c2e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T21:22:44Z","title_canon_sha256":"781d73f9f88aec6314f06215d1c3f28eabcbf5781fee067ad0a4ba2e20b8932c"},"schema_version":"1.0","source":{"id":"1502.02693","kind":"arxiv","version":2}},"canonical_sha256":"7e4ed2178a93760bbf5c4a8391ab663a42876b80a70c2f5a470ea1da7f1ff12b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e4ed2178a93760bbf5c4a8391ab663a42876b80a70c2f5a470ea1da7f1ff12b","first_computed_at":"2026-05-18T01:14:45.551909Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:45.551909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/gvenE1kxw3QQtembdwD2TWgWNo1P6piLoISilDJnStCvoMKHrww44+/8dVfnbB2+BfFFGx8J6pzYE4ynEW0BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:45.552643Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.02693","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66a8f3ee6d9a315226b96a23287a44afbf6b392a94eb1c110eca10c3d5de7f85","sha256:a2a3ecfa1215d30193af7b5ac0e02b15d4ed9598329122e6b2e891e4b368ad19"],"state_sha256":"27d4a7610c5881861f75106b17147faad455992ecc30a0bef5bdd8273306cd0c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMoftUxssdf+KLGcGqpIgdSrFlvLutV/oiY0XSZkIkfD9hTNWX6WGfKVqVSd1LDuwAVr1WQTwG+8r9oymbTRCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T17:37:10.266527Z","bundle_sha256":"d7dd2e2048a3efbc1d89bac217b3d0069f4e89599b4b1023a043bd7659203297"}}