fixed point

This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\\ell_{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.

We present a complete characterization of the metric compactification of $L_p$ spaces for $1 \\leq p

We give a complete description of the horofunction boundary of finite-dimensional $\\ell_{p}$ spaces for $1 \\leq p \\leq \\infty$. We also study the variation norm on $\\mathbf{R}^\\mathcal{N}$, $\\mathcal{N}=\\{1,…,N\\}$, and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert’s projective metric on the interior of the standard cone $\\mathbf{R}^\\mathcal{N}_{+}$ of $\\mathbf{R}^\\mathcal{N}$.