On the action of the group of isometries on a locally compact metric\n space

2009/02/02 by Antonios Manoussos, Manoussos, Antonios · 1 citation
Mathematics · #37B05 #54H20 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.0902.0319

openalex publication_date 2009/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note we give an answer to the following question. Let X be a\nlocally compact metric space with group of isometries G. Let g_i be a\nnet in G for which g_ix converges to y, for some x,y\∈ X. What can we\nsay about the convergence of g_i ? We show that there exist a subnet\n g_j of g_i and an isometry f:C_x\→ X such that g_j converges\nto f pointwise on C_x and f(C_x)=C_f(x), where C_x and C_y denote\nthe pseudo-components of x and y respectively. Applying this we give short\nproofs of the van Dantzig--van der Waerden theorem (1928) and Gao--Kechris\ntheorem (2003).\n

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