2012/09/09 by Oleksiy Dovgoshey, Dovgoshey, Oleksiy, Evgeniy Petrov +1 · 1 citation
Mathematics · #54E25 (Secondary) #54E40 (Primary) 54E35 #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Metric Geometry (math.MG) #math.MG #msc:54E25 #msc:54E35 #msc:54E40
paper · pdf · doi:10.48550/arxiv.1209.1820
23 pages, 1 figure
arxiv created 2012/09/09 · openalex publication_date 2012/09/09 · arxiv updated 2012/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X → Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) → D(X) such that dX = f ∘ dY ∘ F. It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F : X → Y is an isometry if X and Y are ultrametric and compact with D(X) = D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.