{"paper":{"title":"On global linearization of planar involutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Benito Pires, Marco Antonio Teixeira","submitted_at":"2011-05-24T20:55:49Z","abstract_excerpt":"Let $\\phi:\\R^2\\to\\R^2$ be an orientation--preserving $C^1$ involution such that $\\phi(0)=0$ and let ${\\rm Spc}\\,(\\phi)=\\{{\\rm Eigenvalues\\,\\,of}\\,\\, D\\phi(p)\\mid p\\in\\R^2\\}$. We prove that if ${\\rm Spc}\\,{(\\phi)}\\subset\\R$ or ${\\rm Spc}\\,(\\phi)\\cap [1,1+\\epsilon)=\\emptyset$ for some $\\epsilon>0$ then $\\phi$ is globally $C^1$ conjugate to the linear involution $D\\phi(0)$ via the conjugacy $h=(I+D\\phi(0)\\phi)/2$, where $I:\\R^2\\to\\R^2$ is the identity map. Similarly, if $\\phi$ is an orientation-reversing $C^1$ involution such that $\\phi(0)=0$ and ${\\rm Trace}\\,\\big(D\\phi(0)D\\phi(p)\\big)>-1 $ for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4890","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"}