A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains

Abstract

AbstractFor 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.