A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains
Brian Lins · Mathematical Proceedings of the Cambridge Philosophical Society · 2007
Abstract For a polyhedral domain $\Sigma \subset \mathbb{R}^n$ , and a Hilbert metric nonexpansive mapT:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any pointx∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone $\mathbb{R}^n_+$ which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.