The structure of $\omega $-limit sets of nonexpansive maps

Abstract

It is shown that w-limit sets of nonexpansive maps carry the structure of monothetic topological groups. This fact is then used to answer a question of Edelstein. Let T: D -* D be a map from a subset D of a metric space X into itself. We call T nonexpansive if d(T(x), T(y)) C be nonexpansive. If for some x E C the w-limit set G of x is nonempty, then there exists a binary operation in G under which it is a monothetic topological group in the topology induced by the metric of X. (Recall a topological group G is called monothetic if it contains an element x such that {X: n E Z} is dense in G.) PROOF. If G is nonempty, then as before it is minimal and strongly invariant under T, and Tn is an isometric homeomorphism of G for all n (see [4, Theorem 1]). Choose e E G arbitrarily and define a binary operation on G (denoted by juxtaposition) as follows: for any b E G, find a subsequence Nb of Z+ such that limN, Tn(e) = b; then for a E G define ab = limNb Tn(a). Using the fact that T is isometric on G, it is straightforward to show that this limit exists and is unique. In fact, by applying a theorem of Moore [7, p. 100] on double sequences, it can be shown that this operation is abelian. Now the element e is clearly the identity, and since T is a homeomorphism on G, inverses are given by the formula b-'= limNb T-(e). Associativity follows directly from the definition. Thus G is a group. Received by the editors August 17, 1979 and, in revised form, January 3, 1980. AMS (MOS) subject classifications (1970). Primary 47H10.