2024/07/04 by Ravasini, Davide, Thimm, Daylen K.
#46C05 #54E52 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2407.03881
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\inN(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\inN(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\inN(C) and the convergence of the iterates of f to its fixed point.