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

Abstract

Let K be a closed cone with nonempty interior in a Banach space X. Suppose that f : int K → int K 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 ∈ int K . 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.