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For every odd $p\\ge 3$, every $d\\ge 0$ and every $\\lambda\\ge\\lambda_p$, we build a piecewise monotone continuous interval map that is of type $2^dp$ for Sharkovskii's order and whose topological entropy is $\\frac{\\log\\lambda}{2"},"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":"1906.03649","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-06-09T14:31:20Z","cross_cats_sorted":[],"title_canon_sha256":"82da769e38a2c320c0cf1843f26879dfd9fd18551ae9ed872e5672c7c2896aab","abstract_canon_sha256":"59a870f5642642251992c7591f83d1b8d272992e5b9c8dc2d3480a64ddbb2c10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:45.503200Z","signature_b64":"nmcJyVshzepnl2vyWhpDgysF4G1xIKKvUFvvpXUj7UP/tOuDl8Fq4IaBT//rSh765EsGJc+ycSJwv2qlOqiCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2359c7dea45b9e03ae874939b05e4102e2d099485dffd65cc9778a63f9236d09","last_reissued_at":"2026-05-17T23:43:45.502685Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:45.502685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Interval maps of given topological entropy and Sharkovskii's type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sylvie Ruette","submitted_at":"2019-06-09T14:31:20Z","abstract_excerpt":"It is known that the topological entropy of a continuous interval map $f$ is positive if and only if the type of $f$ for Sharkovskii's order is $2^d p$ for some odd integer $p\\ge 3$ and some $d\\ge 0$; and in this case the topological entropy of $f$ is greater than or equal to $\\frac{\\log\\lambda_p}{2^d}$, where $\\lambda_p$ is the unique positive root of $X^p-2X^{p-2}-1$. For every odd $p\\ge 3$, every $d\\ge 0$ and every $\\lambda\\ge\\lambda_p$, we build a piecewise monotone continuous interval map that is of type $2^dp$ for Sharkovskii's order and whose topological entropy is $\\frac{\\log\\lambda}{2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03649","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":"1906.03649","created_at":"2026-05-17T23:43:45.502768+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.03649v1","created_at":"2026-05-17T23:43:45.502768+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.03649","created_at":"2026-05-17T23:43:45.502768+00:00"},{"alias_kind":"pith_short_12","alias_value":"ENM4PXVELOPA","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"ENM4PXVELOPAHLUH","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"ENM4PXVE","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/ENM4PXVELOPAHLUHJE43AXSBAL","json":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL.json","graph_json":"https://pith.science/api/pith-number/ENM4PXVELOPAHLUHJE43AXSBAL/graph.json","events_json":"https://pith.science/api/pith-number/ENM4PXVELOPAHLUHJE43AXSBAL/events.json","paper":"https://pith.science/paper/ENM4PXVE"},"agent_actions":{"view_html":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL","download_json":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL.json","view_paper":"https://pith.science/paper/ENM4PXVE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.03649&json=true","fetch_graph":"https://pith.science/api/pith-number/ENM4PXVELOPAHLUHJE43AXSBAL/graph.json","fetch_events":"https://pith.science/api/pith-number/ENM4PXVELOPAHLUHJE43AXSBAL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL/action/storage_attestation","attest_author":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL/action/author_attestation","sign_citation":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL/action/citation_signature","submit_replication":"https://pith.science/pith/ENM4PXVELOPAHLUHJE43AXSBAL/action/replication_record"}},"created_at":"2026-05-17T23:43:45.502768+00:00","updated_at":"2026-05-17T23:43:45.502768+00:00"}