On selfmaps of metric spaces

Abstract

In this work several questions involving selfmaps of metric and Banach spaces are studied. The converse to a theorem of D.F. Bailey is proved; more precisely, if f is a continuous selfmap of a locally compact metric space X with a fixed point u such that u ε L(x), for all x ε X, then each pair of points in X is proximal. The condition of equicontinuous iterates is introduced as a generalization of nonexpansive selfmaps and it is proved that if f has equicontinuous iterates on X and if x, y ε X, then either L(x) = L(y) or L(x) [intersection] L(y) = 0; thus the relation R = {(x, y) : L(x) = L(y)} is an equivalence relation and the quotient space X/R is studied. In particular, X/R is found to be Hausdorff provided X is compact. Under the same conditions, the sets [intersection] f[superscript n](X), [union][xεX underneath symbol] L(x) and F(f) are compared and sufficient conditions are provided to ensure they be the same set. Property R (for repel) is motivated and introduced as a generalization of diminishing orbital diameters. The existence of a fixed point for maps with property R defined on compact metric spaces is proved. It is further shown that a map with property R and equicontinuous iterates must have the property that z ε L(x) for some x implies f(z) = z. ...