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We assume $\\tau_{1}$, $\\tau_{2}$ to be piecewise monotonic and preserving continuous invariant measures $\\mu_{1}$ and $\\mu_{2}$, respectively. Let $% F^{(1)}$ and $F^{(2)}$ be the distribution functions of $\\mu_{1}$ and $\\mu_{2}.$ The main results shows that for any convex combination $F$ of $% F^{(1)} "},"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":"1309.6009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-23T23:35:47Z","cross_cats_sorted":[],"title_canon_sha256":"fdde445854b4f90b4768e047b911495e2a1cb55d5ec49e389693af76d27b4c72","abstract_canon_sha256":"ca57220c520739491b0920728fb88e174f0d0ad115e5d75a49954ed72d32d5ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:24.391010Z","signature_b64":"O8/yt2EqgstaIH+Rz6Ynzg0lVhJtOswBrVYZSvgopyyzz03SBIEc9eT9rrQw8dMa2c8uR5LYY2+PiW1dpzmBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"609680474278d665715367f1fafed46bd6432fff98bed474f5a1c8b9bf1b3d94","last_reissued_at":"2026-05-18T03:12:24.390281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:24.390281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Selections and their Absolutely Continuous Invariant Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"A. Boyarsky, P. G\\'ora, Zh. Li","submitted_at":"2013-09-23T23:35:47Z","abstract_excerpt":"Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $% I\\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects onto $I_{i.}$ We define the lower and upper maps $\\tau_{1},$ $\\tau_{2}$ by the lower and upper boundaries of $G_{i},i=1,....,n,$ respectively. We assume $\\tau_{1}$, $\\tau_{2}$ to be piecewise monotonic and preserving continuous invariant measures $\\mu_{1}$ and $\\mu_{2}$, respectively. Let $% F^{(1)}$ and $F^{(2)}$ be the distribution functions of $\\mu_{1}$ and $\\mu_{2}.$ The main results shows that for any convex combination $F$ of $% F^{(1)} "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6009","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":"1309.6009","created_at":"2026-05-18T03:12:24.390401+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6009v1","created_at":"2026-05-18T03:12:24.390401+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6009","created_at":"2026-05-18T03:12:24.390401+00:00"},{"alias_kind":"pith_short_12","alias_value":"MCLIAR2CPDLG","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MCLIAR2CPDLGK4KT","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MCLIAR2C","created_at":"2026-05-18T12:27:52.871228+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/MCLIAR2CPDLGK4KTM7Y7V7WUNP","json":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP.json","graph_json":"https://pith.science/api/pith-number/MCLIAR2CPDLGK4KTM7Y7V7WUNP/graph.json","events_json":"https://pith.science/api/pith-number/MCLIAR2CPDLGK4KTM7Y7V7WUNP/events.json","paper":"https://pith.science/paper/MCLIAR2C"},"agent_actions":{"view_html":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP","download_json":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP.json","view_paper":"https://pith.science/paper/MCLIAR2C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6009&json=true","fetch_graph":"https://pith.science/api/pith-number/MCLIAR2CPDLGK4KTM7Y7V7WUNP/graph.json","fetch_events":"https://pith.science/api/pith-number/MCLIAR2CPDLGK4KTM7Y7V7WUNP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP/action/storage_attestation","attest_author":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP/action/author_attestation","sign_citation":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP/action/citation_signature","submit_replication":"https://pith.science/pith/MCLIAR2CPDLGK4KTM7Y7V7WUNP/action/replication_record"}},"created_at":"2026-05-18T03:12:24.390401+00:00","updated_at":"2026-05-18T03:12:24.390401+00:00"}