2006/08/21 by Tim Austin, Tim D. Austin, Austin, Tim D. · 1 citation
Mathematics · #54E40 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #Metric Geometry (math.MG) #math.CA #math.MG #msc:54E40
paper · pdf · doi:10.48550/arxiv.math/0608523
16 pages, 3 postscript figures, to appear in Mathematika
arxiv created 2006/08/21 · openalex publication_date 2006/08/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we disprove the following conjectured generalization of the Contraction Mapping Theorem (due to J.D. Stein Jr.): Let X be a complete metric space and let F be a finite family of self-maps of X. Suppose there is a postive constant strictly less than 1 such that, for any two points x and y of X, some member of F contracts those points by a factor of at most that constant. Then some composition of members of F has a fixed point. We also show that the above does hold for a (continuous) commuting F containing only two maps. We conjecture that it holds for commuting F of any finite size.