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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1506.06872","created_at":"2026-05-18T01:37:22.889220+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.06872v1","created_at":"2026-05-18T01:37:22.889220+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06872","created_at":"2026-05-18T01:37:22.889220+00:00"},{"alias_kind":"pith_short_12","alias_value":"HC64SKJFJTNT","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"HC64SKJFJTNTD43F","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"HC64SKJF","created_at":"2026-05-18T12:29:22.688609+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM","json":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM.json","graph_json":"https://pith.science/api/pith-number/HC64SKJFJTNTD43FPJXYLOBBDM/graph.json","events_json":"https://pith.science/api/pith-number/HC64SKJFJTNTD43FPJXYLOBBDM/events.json","paper":"https://pith.science/paper/HC64SKJF"},"agent_actions":{"view_html":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM","download_json":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM.json","view_paper":"https://pith.science/paper/HC64SKJF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.06872&json=true","fetch_graph":"https://pith.science/api/pith-number/HC64SKJFJTNTD43FPJXYLOBBDM/graph.json","fetch_events":"https://pith.science/api/pith-number/HC64SKJFJTNTD43FPJXYLOBBDM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/action/storage_attestation","attest_author":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/action/author_attestation","sign_citation":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/action/citation_signature","submit_replication":"https://pith.science/pith/HC64SKJFJTNTD43FPJXYLOBBDM/action/replication_record"}},"created_at":"2026-05-18T01:37:22.889220+00:00","updated_at":"2026-05-18T01:37:22.889220+00:00"}