On the Action of the Group of Isometries on a Locally Compact Metric Space: Closed-open Partitions and Closed Orbits
نویسنده
چکیده
In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closed-open subsets of X . Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact with respect to the compactopen topology. In case G is locally compact we decompose the space X into closed-open invariant disjoint sets that are related to various limit behaviors of the orbits. We also provide a simple example of a locally compact separable and complete metric space X with discrete group of isometries G such that the natural action of G on X has closed and non-closed orbits.
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