{"paper":{"title":"Fixed points in non-invariant plane continua","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"AL, Alexander Blokh, Birmingham, Lex Oversteegen (UAB, USA)","submitted_at":"2008-05-08T15:56:06Z","abstract_excerpt":"If $f:[a,b]\\to \\mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:\\mathbb{C}\\to\\mathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $X\\to D, X\\subset D$ and for positively oriented maps $f:X\\to \\mathbb{C}, X\\subset \\mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:\\mathbb{C}\\to\\mathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $a\\in \\mathrm{Int}(X)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0805.1069","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":""},"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"}