{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:FVU4WS4ZQUMICSRSYC2KOT7OPR","short_pith_number":"pith:FVU4WS4Z","schema_version":"1.0","canonical_sha256":"2d69cb4b998518814a32c0b4a74fee7c7bf84bbaa0f708c607e3b22d22752d8b","source":{"kind":"arxiv","id":"1901.01064","version":1},"attestation_state":"computed","paper":{"title":"Dense chaos for continuous interval maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sylvie Ruette","submitted_at":"2019-01-04T11:47:24Z","abstract_excerpt":"A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that $\\limsup_{n\\to+\\infty}|f^n(x)-f^n(y)|>0$ and $\\liminf_{n\\to+\\infty} |f^n(x)-f^n(y)|=0$ is dense (resp. residual) in $I\\times I$. We prove that if the interval map $f$ is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which containing a horseshoe for $f^2$. It implies that every densely chaotic interval map is of type at most $6$ for Sharkovsky's order (that is, there exists a periodic point of period "},"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":"1901.01064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-04T11:47:24Z","cross_cats_sorted":[],"title_canon_sha256":"7fa61b6e32b782043b91fd8005ea950e8ce10cf1ab9da36c9f0ce28f82977f2d","abstract_canon_sha256":"f12fe12957095a36eebbc3f243ea508648a700da2c3b7f9617eb6bcdbec8f0e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:44.339369Z","signature_b64":"yVjw003mGGRTJfx3GxPlFswdfcFbaxkjEI3l+MAnxcJuYTezckmZXvOz7pVE/CdeY1wgxBRax1cHjRxSN3sOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d69cb4b998518814a32c0b4a74fee7c7bf84bbaa0f708c607e3b22d22752d8b","last_reissued_at":"2026-05-17T23:56:44.338959Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:44.338959Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dense chaos for continuous interval maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sylvie Ruette","submitted_at":"2019-01-04T11:47:24Z","abstract_excerpt":"A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that $\\limsup_{n\\to+\\infty}|f^n(x)-f^n(y)|>0$ and $\\liminf_{n\\to+\\infty} |f^n(x)-f^n(y)|=0$ is dense (resp. residual) in $I\\times I$. We prove that if the interval map $f$ is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which containing a horseshoe for $f^2$. It implies that every densely chaotic interval map is of type at most $6$ for Sharkovsky's order (that is, there exists a periodic point of period "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01064","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":"1901.01064","created_at":"2026-05-17T23:56:44.339023+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.01064v1","created_at":"2026-05-17T23:56:44.339023+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01064","created_at":"2026-05-17T23:56:44.339023+00:00"},{"alias_kind":"pith_short_12","alias_value":"FVU4WS4ZQUMI","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"FVU4WS4ZQUMICSRS","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"FVU4WS4Z","created_at":"2026-05-18T12:33:15.570797+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/FVU4WS4ZQUMICSRSYC2KOT7OPR","json":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR.json","graph_json":"https://pith.science/api/pith-number/FVU4WS4ZQUMICSRSYC2KOT7OPR/graph.json","events_json":"https://pith.science/api/pith-number/FVU4WS4ZQUMICSRSYC2KOT7OPR/events.json","paper":"https://pith.science/paper/FVU4WS4Z"},"agent_actions":{"view_html":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR","download_json":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR.json","view_paper":"https://pith.science/paper/FVU4WS4Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.01064&json=true","fetch_graph":"https://pith.science/api/pith-number/FVU4WS4ZQUMICSRSYC2KOT7OPR/graph.json","fetch_events":"https://pith.science/api/pith-number/FVU4WS4ZQUMICSRSYC2KOT7OPR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR/action/storage_attestation","attest_author":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR/action/author_attestation","sign_citation":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR/action/citation_signature","submit_replication":"https://pith.science/pith/FVU4WS4ZQUMICSRSYC2KOT7OPR/action/replication_record"}},"created_at":"2026-05-17T23:56:44.339023+00:00","updated_at":"2026-05-17T23:56:44.339023+00:00"}