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 f : intK →intK is order-preserving and homogeneous of degree one. Let q :K → [0,∞) 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. © 2008 Elsevier Inc. All rights reserved.