Generic nonexpansive Hilbert space mappings

We consider a closed convex set C in a separable, infinite-dimensional Hilbert space and endow the set N(C) of nonexpansive self-mappings on C with the topology of pointwise convergence. We introduce the notion of a somewhat bounded set and establish a strong connection between this property and the existence of fixed points for the generic f(C), in the sense of Baire categories. Namely, if C is somewhat bounded, the generic nonexpansive mapping on C admits a fixed point, whereas if C is not somewhat bounded, the generic nonexpansive mapping on C does not have any fixed points. This results in a topological 0-1 law: the set of all f(C) with a fixed point is either meager or residual. We further prove that, generically, there are no fixed points in the interior of C and, under additional geometric assumptions, we show the uniqueness of such fixed points for the generic f(C) and the convergence of the iterates of f to its fixed point.