Generic nonexpansive Hilbert space mappings

We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal N(C)$ 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 $f\in \mathcal N(C)$, in the sense of Baire category. 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 $f\in \mathcal N(C)$ 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 $f\in \mathcal N(C)$ and the convergence of the iterates of $f$ to its fixed point.