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

For a polyhedral domain Sigma subset of R(n), and a Hilbert metric nonexpansive map T: Sigma --> Sigma which does not have a fixed point in Sigma, we prove that the omega limit set omega(x; T) of any point x is an element of Sigma is contained in a convex subset of the boundary partial derivative Sigma. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone 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.