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The map $x\\mapsto \\beta x-\\lfloor \\beta x\\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\\mathscr{P}u=u$ and $\\mathscr{P}v=\\beta^{-1} v$ such that for ev"},"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":"2502.17113","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-24T12:53:17Z","cross_cats_sorted":[],"title_canon_sha256":"3a6a13127a5716b88f5b5927c1c10a27dd60f7dc32f149e09495d057c0ce1174","abstract_canon_sha256":"ddd98ec4317403f05024ae141a0f638a7be28914c0362b2613b1e221fc91f342"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:04:52.169096Z","signature_b64":"76nTXe9bFUjshbr3Gz21QFdflLqmwvjcmt6sH2IUCoArycbzoPH7bhZo7NY8t8JcRCmG24XLvSIbABpPJ9mWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4771abb00c1094d2850749331ff91b19db63e9542e1992461d73f10caaf2e5b7","last_reissued_at":"2026-05-20T01:04:52.168136Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:04:52.168136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp iteration asymptotics for transfer operators induced by greedy $\\beta$-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Horia D. Cornean, Kasper S. S{\\o}rensen","submitted_at":"2025-02-24T12:53:17Z","abstract_excerpt":"We consider base-$\\beta$ expansions of Parry's type, where $a_0 \\geq a_1 \\geq 1$ are integers and $a_0<\\beta <a_0+1$ is the positive solution to $\\beta^2 = a_0\\beta + a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\\mapsto \\beta x-\\lfloor \\beta x\\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\\mathscr{P}u=u$ and $\\mathscr{P}v=\\beta^{-1} v$ such that for ev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.17113","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.17113/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2502.17113","created_at":"2026-05-20T01:04:52.168262+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.17113v2","created_at":"2026-05-20T01:04:52.168262+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.17113","created_at":"2026-05-20T01:04:52.168262+00:00"},{"alias_kind":"pith_short_12","alias_value":"I5Y2XMAMCCKN","created_at":"2026-05-20T01:04:52.168262+00:00"},{"alias_kind":"pith_short_16","alias_value":"I5Y2XMAMCCKNFBIH","created_at":"2026-05-20T01:04:52.168262+00:00"},{"alias_kind":"pith_short_8","alias_value":"I5Y2XMAM","created_at":"2026-05-20T01:04:52.168262+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/I5Y2XMAMCCKNFBIHJEZR76I3DH","json":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH.json","graph_json":"https://pith.science/api/pith-number/I5Y2XMAMCCKNFBIHJEZR76I3DH/graph.json","events_json":"https://pith.science/api/pith-number/I5Y2XMAMCCKNFBIHJEZR76I3DH/events.json","paper":"https://pith.science/paper/I5Y2XMAM"},"agent_actions":{"view_html":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH","download_json":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH.json","view_paper":"https://pith.science/paper/I5Y2XMAM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.17113&json=true","fetch_graph":"https://pith.science/api/pith-number/I5Y2XMAMCCKNFBIHJEZR76I3DH/graph.json","fetch_events":"https://pith.science/api/pith-number/I5Y2XMAMCCKNFBIHJEZR76I3DH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH/action/storage_attestation","attest_author":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH/action/author_attestation","sign_citation":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH/action/citation_signature","submit_replication":"https://pith.science/pith/I5Y2XMAMCCKNFBIHJEZR76I3DH/action/replication_record"}},"created_at":"2026-05-20T01:04:52.168262+00:00","updated_at":"2026-05-20T01:04:52.168262+00:00"}