{"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. We also prove that every injective piecewise continuous interval map with no connections and no "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02542","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"}