{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:HC64SKJFJTNTD43FPJXYLOBBDM","short_pith_number":"pith:HC64SKJF","canonical_record":{"source":{"id":"1506.06872","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-23T06:14:09Z","cross_cats_sorted":[],"title_canon_sha256":"a4ebdce4a1685d7739c709d27d856e1978ce2d2c0d14102d0c9be2e28fde66be","abstract_canon_sha256":"96cacca7a8220eedcf35624be744477f8b8f30284098057eaf1721e7e41b3880"},"schema_version":"1.0"},"canonical_sha256":"38bdc929254cdb31f3657a6f85b8211b33d1d0e6ec28cc562f0d28eef1758388","source":{"kind":"arxiv","id":"1506.06872","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06872","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06872v1","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06872","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"pith_short_12","alias_value":"HC64SKJFJTNT","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"HC64SKJFJTNTD43F","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"HC64SKJF","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:HC64SKJFJTNTD43FPJXYLOBBDM","target":"record","payload":{"canonical_record":{"source":{"id":"1506.06872","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-23T06:14:09Z","cross_cats_sorted":[],"title_canon_sha256":"a4ebdce4a1685d7739c709d27d856e1978ce2d2c0d14102d0c9be2e28fde66be","abstract_canon_sha256":"96cacca7a8220eedcf35624be744477f8b8f30284098057eaf1721e7e41b3880"},"schema_version":"1.0"},"canonical_sha256":"38bdc929254cdb31f3657a6f85b8211b33d1d0e6ec28cc562f0d28eef1758388","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:22.889837Z","signature_b64":"S+EzMGTVnKxEf7p3Nd8UxjONZnhyvLkqWJxr12CS+Bv7h/TrBNlq+j7WjrqxBra1xNSRymPFtTJNN0SpQWE3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38bdc929254cdb31f3657a6f85b8211b33d1d0e6ec28cc562f0d28eef1758388","last_reissued_at":"2026-05-18T01:37:22.889107Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:22.889107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.06872","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":"2026-05-18T01:37:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M+GXRjutpyVQvwiSVplj5pJPUy9lKUFEzoxZdoisyHHU5FGBt3KbOzi4bc8j45Ise7cG1oAxPNuuIrLkv9HiAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:46:21.013066Z"},"content_sha256":"76580ff711c3be7a86dd8d8f2835449eeaffdfd86575fe91328979ebf7c107b7","schema_version":"1.0","event_id":"sha256:76580ff711c3be7a86dd8d8f2835449eeaffdfd86575fe91328979ebf7c107b7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:HC64SKJFJTNTD43FPJXYLOBBDM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Li-Yorke chaos for dendrite maps with zero topological entropy and $\\omega$-limit sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ghassen Askri","submitted_at":"2015-06-23T06:14:09Z","abstract_excerpt":"Let $X$ be a dendrite  with  set of endpoints $E(X)$ closed  and let $f:~X \\to X$ be a continuous map with zero topological entropy.  Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\\omega$-limit set of $f$ then $L\\cap P(f)\\subset E(X)^{\\prime}$, where $E(X)^{\\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\\cap P(f)=\\emptyset$. We also show  that if $E(X)^{\\prime}$ is finite then any uncountable $\\omega$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06872","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":"2026-05-18T01:37:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NunmqzMDbGUmTzDpLbC3mdRC6R07gyW4v2Vhu5iVordIHDnQ2VlX6HEEFn5443JdWlQIXNS3Zw2O4vuYzG/1AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:46:21.013431Z"},"content_sha256":"0ba8a97b20dd2d14335d4d06b4a9739a9c0143934caa5fe42538d8f74de46377","schema_version":"1.0","event_id":"sha256:0ba8a97b20dd2d14335d4d06b4a9739a9c0143934caa5fe42538d8f74de46377"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/bundle.json","state_url":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HC64SKJFJTNTD43FPJXYLOBBDM/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":"2026-06-21T21:46:21Z","links":{"resolver":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM","bundle":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/bundle.json","state":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HC64SKJFJTNTD43FPJXYLOBBDM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HC64SKJFJTNTD43FPJXYLOBBDM","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":"96cacca7a8220eedcf35624be744477f8b8f30284098057eaf1721e7e41b3880","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-23T06:14:09Z","title_canon_sha256":"a4ebdce4a1685d7739c709d27d856e1978ce2d2c0d14102d0c9be2e28fde66be"},"schema_version":"1.0","source":{"id":"1506.06872","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06872","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06872v1","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06872","created_at":"2026-05-18T01:37:22Z"},{"alias_kind":"pith_short_12","alias_value":"HC64SKJFJTNT","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"HC64SKJFJTNTD43F","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"HC64SKJF","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:0ba8a97b20dd2d14335d4d06b4a9739a9c0143934caa5fe42538d8f74de46377","target":"graph","created_at":"2026-05-18T01:37:22Z","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":"Let $X$ be a dendrite  with  set of endpoints $E(X)$ closed  and let $f:~X \\to X$ be a continuous map with zero topological entropy.  Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\\omega$-limit set of $f$ then $L\\cap P(f)\\subset E(X)^{\\prime}$, where $E(X)^{\\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\\cap P(f)=\\emptyset$. We also show  that if $E(X)^{\\prime}$ is finite then any uncountable $\\omega$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yor","authors_text":"Ghassen Askri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-23T06:14:09Z","title":"Li-Yorke chaos for dendrite maps with zero topological entropy and $\\omega$-limit sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06872","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":"sha256:76580ff711c3be7a86dd8d8f2835449eeaffdfd86575fe91328979ebf7c107b7","target":"record","created_at":"2026-05-18T01:37:22Z","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":"96cacca7a8220eedcf35624be744477f8b8f30284098057eaf1721e7e41b3880","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-23T06:14:09Z","title_canon_sha256":"a4ebdce4a1685d7739c709d27d856e1978ce2d2c0d14102d0c9be2e28fde66be"},"schema_version":"1.0","source":{"id":"1506.06872","kind":"arxiv","version":1}},"canonical_sha256":"38bdc929254cdb31f3657a6f85b8211b33d1d0e6ec28cc562f0d28eef1758388","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38bdc929254cdb31f3657a6f85b8211b33d1d0e6ec28cc562f0d28eef1758388","first_computed_at":"2026-05-18T01:37:22.889107Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:22.889107Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S+EzMGTVnKxEf7p3Nd8UxjONZnhyvLkqWJxr12CS+Bv7h/TrBNlq+j7WjrqxBra1xNSRymPFtTJNN0SpQWE3Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:22.889837Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.06872","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76580ff711c3be7a86dd8d8f2835449eeaffdfd86575fe91328979ebf7c107b7","sha256:0ba8a97b20dd2d14335d4d06b4a9739a9c0143934caa5fe42538d8f74de46377"],"state_sha256":"e6a15661bd0e92258821fd8147eb12a2c3c0e4003c9cbf37c33478a2754297c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hi4NZiAziOsY2Kf9JxmoPFn0PS9QlTAZ46qQuuINeNQzhVLN78GVNHFpci4qIIz21LnmGa1cVZbfgHu1UwqrAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T21:46:21.015250Z","bundle_sha256":"7213fdf37a8318fa650c0d9d8e7e8ee6839bab2eac69e792a5888c1154d9d279"}}