Iterated linear maps on a cone and Denjoy–Wolff theorems

Let C be a closed cone with nonempty interior int(C) in a finite dimensional Banach space X. We consider linear maps f : X → X such that f(int(C)) ⊂ int(C) and f has no eigenvector in int(C). For q ∈ C∗, with q(x) > 0 ∀x ∈ C⧹{0} we define and Σq = {x ∈ C∣q(x) = 1}. Let ri(Σq) denote the relative interior of Σq. We are interested in the omega limit set ω(x; T) of x ∈ ri(Σq) under T. We prove that the convex hull co(ω(x; T)) ⊂ ∂Σq, and if C is polyhedral we also show that ω(x; T) is finite. Thus if C is polyhedral there is a face of C such that the orbit of any point in the interior of C under iterates of f approaches that face after scaling.