{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:KPDRWNED5JTPHZJUE2ZK2TBTNJ","short_pith_number":"pith:KPDRWNED","canonical_record":{"source":{"id":"1709.00527","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-09-02T03:23:57Z","cross_cats_sorted":[],"title_canon_sha256":"50bf7ab4bfc17f7881a5905625764c105e788d744cc32978ca8ddabb740188d7","abstract_canon_sha256":"1ae2c4a15fcf022d89f1a2ed4b58c4b1425000f3d4813b5aed707c467eb6a345"},"schema_version":"1.0"},"canonical_sha256":"53c71b3483ea66f3e53426b2ad4c336a7a64921966ed07979fdfbd57a9e32413","source":{"kind":"arxiv","id":"1709.00527","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00527","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00527v2","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00527","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"pith_short_12","alias_value":"KPDRWNED5JTP","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"KPDRWNED5JTPHZJU","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"KPDRWNED","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:KPDRWNED5JTPHZJUE2ZK2TBTNJ","target":"record","payload":{"canonical_record":{"source":{"id":"1709.00527","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-09-02T03:23:57Z","cross_cats_sorted":[],"title_canon_sha256":"50bf7ab4bfc17f7881a5905625764c105e788d744cc32978ca8ddabb740188d7","abstract_canon_sha256":"1ae2c4a15fcf022d89f1a2ed4b58c4b1425000f3d4813b5aed707c467eb6a345"},"schema_version":"1.0"},"canonical_sha256":"53c71b3483ea66f3e53426b2ad4c336a7a64921966ed07979fdfbd57a9e32413","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:17.975343Z","signature_b64":"ltVa6lUhQcIWoqKzHowrY7sGih5fG4L/K/3qOuroHkDR7P6TVm0dROtllMzzc4BvS+DlundbH2e7kqfkHIXqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53c71b3483ea66f3e53426b2ad4c336a7a64921966ed07979fdfbd57a9e32413","last_reissued_at":"2026-05-17T23:49:17.974761Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:17.974761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.00527","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":"2026-05-17T23:49:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XkmBiqV+F085Fhj6lkJxOmywVu9TACWf9HvySiIKQhb9+IrwC1j81/TTgS+82h69UJNsmvvTli9tm5B05QLEAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T15:12:02.321380Z"},"content_sha256":"641b2cea0093f2d84fa2caf242453dd61ae04e7576dbf4f10b6ae07608f46d16","schema_version":"1.0","event_id":"sha256:641b2cea0093f2d84fa2caf242453dd61ae04e7576dbf4f10b6ae07608f46d16"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:KPDRWNED5JTPHZJUE2ZK2TBTNJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Fang Wang, Hong-Kun Zhang, Jianyu Chen","submitted_at":"2017-09-02T03:23:57Z","abstract_excerpt":"For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00527","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":"2026-05-17T23:49:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RvyBvD14b2ioxVdeBDN4g8Ih8I9BEli67i/RGydMBm/zhbNZ+o108AOghZEvSyEsPWFfl4qvpA4KgmZafINsBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T15:12:02.321729Z"},"content_sha256":"278ef20045747e7854a8e8e52bbb8cd186b97b48ed3b9d8bfd6960589195336d","schema_version":"1.0","event_id":"sha256:278ef20045747e7854a8e8e52bbb8cd186b97b48ed3b9d8bfd6960589195336d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/bundle.json","state_url":"https://pith.science/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/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":"2026-05-20T15:12:02Z","links":{"resolver":"https://pith.science/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ","bundle":"https://pith.science/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/bundle.json","state":"https://pith.science/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KPDRWNED5JTPHZJUE2ZK2TBTNJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:KPDRWNED5JTPHZJUE2ZK2TBTNJ","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":"1ae2c4a15fcf022d89f1a2ed4b58c4b1425000f3d4813b5aed707c467eb6a345","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-09-02T03:23:57Z","title_canon_sha256":"50bf7ab4bfc17f7881a5905625764c105e788d744cc32978ca8ddabb740188d7"},"schema_version":"1.0","source":{"id":"1709.00527","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.00527","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"arxiv_version","alias_value":"1709.00527v2","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00527","created_at":"2026-05-17T23:49:17Z"},{"alias_kind":"pith_short_12","alias_value":"KPDRWNED5JTP","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"KPDRWNED5JTPHZJU","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"KPDRWNED","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:278ef20045747e7854a8e8e52bbb8cd186b97b48ed3b9d8bfd6960589195336d","target":"graph","created_at":"2026-05-17T23:49:17Z","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":"For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentia","authors_text":"Fang Wang, Hong-Kun Zhang, Jianyu Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-09-02T03:23:57Z","title":"Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00527","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":"sha256:641b2cea0093f2d84fa2caf242453dd61ae04e7576dbf4f10b6ae07608f46d16","target":"record","created_at":"2026-05-17T23:49:17Z","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":"1ae2c4a15fcf022d89f1a2ed4b58c4b1425000f3d4813b5aed707c467eb6a345","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-09-02T03:23:57Z","title_canon_sha256":"50bf7ab4bfc17f7881a5905625764c105e788d744cc32978ca8ddabb740188d7"},"schema_version":"1.0","source":{"id":"1709.00527","kind":"arxiv","version":2}},"canonical_sha256":"53c71b3483ea66f3e53426b2ad4c336a7a64921966ed07979fdfbd57a9e32413","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53c71b3483ea66f3e53426b2ad4c336a7a64921966ed07979fdfbd57a9e32413","first_computed_at":"2026-05-17T23:49:17.974761Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:17.974761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ltVa6lUhQcIWoqKzHowrY7sGih5fG4L/K/3qOuroHkDR7P6TVm0dROtllMzzc4BvS+DlundbH2e7kqfkHIXqAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:17.975343Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.00527","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:641b2cea0093f2d84fa2caf242453dd61ae04e7576dbf4f10b6ae07608f46d16","sha256:278ef20045747e7854a8e8e52bbb8cd186b97b48ed3b9d8bfd6960589195336d"],"state_sha256":"e3a0c672c7a9f7ae82315a2ebe413ca651a6392b83cb3aca429b58b9506d2206"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ttc+BQdSwdlz5EhFUaXWzqLQBVYu4+wJ2LTiX5Gptxz4plj9UV/cgmNCJ/BrteGO/ET63w8caImM0PQiWXJhCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T15:12:02.323724Z","bundle_sha256":"4cbf1a05c34e6977cb6951405b304393059f8d371663abdae664a19a754dce2a"}}