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We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no "},"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":"1603.02542","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-03-08T15:09:56Z","cross_cats_sorted":[],"title_canon_sha256":"36346ac57d7c079471d08d6405fc45d1a221fc4c6002fb176adb7298f3265401","abstract_canon_sha256":"907a6a1a4939010dcbbdc7fcd85941c20417169d910a89e2f457c49986d241ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:22.947917Z","signature_b64":"KzvSWwf2/wRUaPuRSoMkYyqzXUlbYbiRZaCtv3ubuNYi+uuAE0EZnliMDDR6jfiykcOXoewSSE06kTW5DLoiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02c8a020e2e1887ecf9c96a2904c6a063cfa447a9b49305a683ae24fd3880fc5","last_reissued_at":"2026-05-18T01:19:22.947214Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:22.947214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant measures for piecewise continuous maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Benito Pires","submitted_at":"2016-03-08T15:09:56Z","abstract_excerpt":"We say that $f:[0,1]\\to [0,1]$ is a {\\it piecewise continuous interval map} if there exists a partition $0=x_0<x_1<\\cdots<x_{d}<x_{d+1}=1$ of $[0,1]$ such that $f\\vert_{(x_{i-1},x_i)}$ is continuous and the lateral limits $w_0^+=\\lim_{x\\to 0^+} f(x)$, $w_{d+1}^-=\\lim_{x\\to 1^-} f(x)$, \\mbox{$w_i^{-}=\\lim_{x\\to x_i^-} f(x)$} and $w_i^{+}=\\lim_{x\\to x_i^+} f(x)$ exist for each $i$. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. 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