Denjoy–Wolff theorems, Hilbert metric nonexpansive maps and reproduction–decimation operators

Let K be a closed cone with nonempty interior in a Banach space X. Suppose that is order-preserving and homogeneous of degree one. Let be a continuous, homogeneous of degree one map such that q(x)>0 for all x∈K∖{0}. Let T(x)=f(x)/q(f(x)). We give conditions on the cone K and the map f which imply that there is a convex subset of ∂K which contains the omega limit set ω(x;T) for every x∈intK. We show that these conditions are satisfied by reproduction–decimation operators. We also prove that ω(x;T)⊂∂K for a class of operator-valued means.