{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:QQOG7ACRZYJTOJK22655LX5ISW","short_pith_number":"pith:QQOG7ACR","canonical_record":{"source":{"id":"2503.18092","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-03-23T14:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"85d5d269734176a92dd2c682740feea7b0f3c86f9f9ffc39c991eb4b61692fa9","abstract_canon_sha256":"f25e67436f57af93ecdf2d405c75950b8216606970a428e50599e195b778fa1a"},"schema_version":"1.0"},"canonical_sha256":"841c6f8051ce1337255ad7bbd5dfa895bc085ddcc3805bb372a8ca46ad93bbc1","source":{"kind":"arxiv","id":"2503.18092","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.18092","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"arxiv_version","alias_value":"2503.18092v2","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.18092","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_12","alias_value":"QQOG7ACRZYJT","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_16","alias_value":"QQOG7ACRZYJTOJK2","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_8","alias_value":"QQOG7ACR","created_at":"2026-07-05T11:14:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:QQOG7ACRZYJTOJK22655LX5ISW","target":"record","payload":{"canonical_record":{"source":{"id":"2503.18092","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-03-23T14:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"85d5d269734176a92dd2c682740feea7b0f3c86f9f9ffc39c991eb4b61692fa9","abstract_canon_sha256":"f25e67436f57af93ecdf2d405c75950b8216606970a428e50599e195b778fa1a"},"schema_version":"1.0"},"canonical_sha256":"841c6f8051ce1337255ad7bbd5dfa895bc085ddcc3805bb372a8ca46ad93bbc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:14:05.921708Z","signature_b64":"ZzuyZ8Gpg2I53qfYmAGU1k+gN3XVhjJJClz3bWN98fhdsRw5C1nkZyJ8xubwu52cCBoWTZad2wF/r1jlXsZgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"841c6f8051ce1337255ad7bbd5dfa895bc085ddcc3805bb372a8ca46ad93bbc1","last_reissued_at":"2026-07-05T11:14:05.921210Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:14:05.921210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2503.18092","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":"2026-07-05T11:14:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XBWkBpIWwp+xVZyeiWO4LyNJkdy9zsxQwAomm333bjIblzdSW8Nord7eTS6DgNW+oSf7+1Uq+dDQg1cbVG/ZCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T06:25:53.364745Z"},"content_sha256":"bf793b31346bdaee430e799716ee2894cc815846f54b060fdea125db995af54e","schema_version":"1.0","event_id":"sha256:bf793b31346bdaee430e799716ee2894cc815846f54b060fdea125db995af54e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:QQOG7ACRZYJTOJK22655LX5ISW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodic Optimization for Open Expanding Multi-valued Topological Dynamical Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Oliver Jenkinson, Xiaoran Li, Yiwei Zhang, Yuexin Liao","submitted_at":"2025-03-23T14:44:25Z","abstract_excerpt":"We study the optimization of ergodic averages for multi-valued dynamical systems, i.e. where points may have multiple different forward orbits. Under upper semi-continuity assumptions, we show that the maximum space average with respect to invariant probability measures for such systems can be characterised in terms of maximum time averages on an auxiliary shift space. For all multi-valued expanding systems that are open mappings, we show that every H\\\"older continuous real-valued function can be modified by a coboundary, of the same H\\\"older exponent, such that the resulting function is domin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.18092","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.18092/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:14:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xIWl3IlFlPIu9noK36jACRm2xCXZT7gy/CKQe6oSuB6fTSEfOqyelAP3A4M+eWF3q4ldpK3uZcuowQJWbt24DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T06:25:53.365237Z"},"content_sha256":"96a728a2dde8b57dbbbf7c4323436c8db63140eede9446fbdc7af8487b3cf2c0","schema_version":"1.0","event_id":"sha256:96a728a2dde8b57dbbbf7c4323436c8db63140eede9446fbdc7af8487b3cf2c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QQOG7ACRZYJTOJK22655LX5ISW/bundle.json","state_url":"https://pith.science/pith/QQOG7ACRZYJTOJK22655LX5ISW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QQOG7ACRZYJTOJK22655LX5ISW/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":"2026-08-08T06:25:53Z","links":{"resolver":"https://pith.science/pith/QQOG7ACRZYJTOJK22655LX5ISW","bundle":"https://pith.science/pith/QQOG7ACRZYJTOJK22655LX5ISW/bundle.json","state":"https://pith.science/pith/QQOG7ACRZYJTOJK22655LX5ISW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QQOG7ACRZYJTOJK22655LX5ISW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:QQOG7ACRZYJTOJK22655LX5ISW","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":"f25e67436f57af93ecdf2d405c75950b8216606970a428e50599e195b778fa1a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-03-23T14:44:25Z","title_canon_sha256":"85d5d269734176a92dd2c682740feea7b0f3c86f9f9ffc39c991eb4b61692fa9"},"schema_version":"1.0","source":{"id":"2503.18092","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.18092","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"arxiv_version","alias_value":"2503.18092v2","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.18092","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_12","alias_value":"QQOG7ACRZYJT","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_16","alias_value":"QQOG7ACRZYJTOJK2","created_at":"2026-07-05T11:14:05Z"},{"alias_kind":"pith_short_8","alias_value":"QQOG7ACR","created_at":"2026-07-05T11:14:05Z"}],"graph_snapshots":[{"event_id":"sha256:96a728a2dde8b57dbbbf7c4323436c8db63140eede9446fbdc7af8487b3cf2c0","target":"graph","created_at":"2026-07-05T11:14:05Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2503.18092/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the optimization of ergodic averages for multi-valued dynamical systems, i.e. where points may have multiple different forward orbits. Under upper semi-continuity assumptions, we show that the maximum space average with respect to invariant probability measures for such systems can be characterised in terms of maximum time averages on an auxiliary shift space. For all multi-valued expanding systems that are open mappings, we show that every H\\\"older continuous real-valued function can be modified by a coboundary, of the same H\\\"older exponent, such that the resulting function is domin","authors_text":"Oliver Jenkinson, Xiaoran Li, Yiwei Zhang, Yuexin Liao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-03-23T14:44:25Z","title":"Ergodic Optimization for Open Expanding Multi-valued Topological Dynamical Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.18092","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:bf793b31346bdaee430e799716ee2894cc815846f54b060fdea125db995af54e","target":"record","created_at":"2026-07-05T11:14:05Z","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":"f25e67436f57af93ecdf2d405c75950b8216606970a428e50599e195b778fa1a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-03-23T14:44:25Z","title_canon_sha256":"85d5d269734176a92dd2c682740feea7b0f3c86f9f9ffc39c991eb4b61692fa9"},"schema_version":"1.0","source":{"id":"2503.18092","kind":"arxiv","version":2}},"canonical_sha256":"841c6f8051ce1337255ad7bbd5dfa895bc085ddcc3805bb372a8ca46ad93bbc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"841c6f8051ce1337255ad7bbd5dfa895bc085ddcc3805bb372a8ca46ad93bbc1","first_computed_at":"2026-07-05T11:14:05.921210Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:14:05.921210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZzuyZ8Gpg2I53qfYmAGU1k+gN3XVhjJJClz3bWN98fhdsRw5C1nkZyJ8xubwu52cCBoWTZad2wF/r1jlXsZgDw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:14:05.921708Z","signed_message":"canonical_sha256_bytes"},"source_id":"2503.18092","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf793b31346bdaee430e799716ee2894cc815846f54b060fdea125db995af54e","sha256:96a728a2dde8b57dbbbf7c4323436c8db63140eede9446fbdc7af8487b3cf2c0"],"state_sha256":"df8742af00f4d42f7d2e30bdf266c8b0cf164518c6f5f6ff4ae2d7ad8edde778"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p3uSQ5YPqi1NXmyH7XxaG6t/qVF0hpZoJu40pTXm4sRO3sMJ4r7BKNI0uLbuY5HJxEVuAz1yAit7nrj0UdZmAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T06:25:53.369161Z","bundle_sha256":"fed386f8f5fa2c5f800f76b57c1d82ff40ce6bfa129bed64fc307109b0bdbced"}}