Comments on the cosmic convergence of nonexpansive maps

TL;DR

This paper explores the long-term behavior of nonexpansive maps, linking it to the invariant subspace problem, and presents new results for maps on ll^{1}, while clarifying inaccuracies in existing literature.

Contribution

It introduces a novel result for nonexpansive maps on ll^{1} and clarifies misconceptions in previous studies using metric functionals.

Findings

New result for nonexpansive maps on ll^{1}

Connection established between asymptotic behavior and invariant subspace problem

Identification of inaccuracies in existing literature

Abstract

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.