The group of isometries of a locally compact metric space with one end

In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space ( X , d ) with one end. Using the notion of pseudo-components introduced by S. Gao and A.S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not compact and G acts properly on this pseudo-component. The complement of the non-compact component is a compact subset of X and G may fail to act properly on it.