{"paper":{"title":"Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Maria Fernanda Barrozo, Ursula Molter","submitted_at":"2013-07-03T17:46:13Z","abstract_excerpt":"We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\\mathcal{F}=\\{F_i:i\\in\\mathbb N\\}$. We show the existence of a {\\em smallest} invariant set (with respect to inclusion) for $\\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\\x) = r_i \\boldmath{x} + b_i$ on $X=\\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. Precisely, we can show that for that case, there exists a {\\em unique} bounded invariant set if and only if $r = \\sup_i r_i$ is strictly smaller than 1.\n  Further, if $\\rho = \\{\\rho_k\\}_{k\\in \\mathbb N}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1090","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"}