Iterated linear maps on a cone and Denjoy–Wolff theorems

Abstract

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 T ( x ) = f ( x ) q ( f ( x ) ) and Σ q = { x ∈ C∣ q( x) = 1}. Let ri( Σ q ) denote the relative interior of Σ q . We are interested in the omega limit set ω( x; T) of x ∈ ri( Σ q ) under T. We prove that the convex hull co( ω( x; T)) ⊂ ∂ Σ q , 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.