We introduce the notion of a firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.
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
The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\\ell_p$ spaces for all $1 \\leq p