Iteration of order preserving subhomogeneous maps on a cone

Abstract

Abstract. We investigate the iterative behaviour of continuous order preserving subhomogeneous maps f : K → K, where K is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of f converges to a periodic orbit and, moreover, the period of each periodic point of f is bounded bywhere N is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in R n , we show that the upper bound is asymptotically sharp.These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in R n .

Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

“…Conjecture 1.1 was shown to hold in case Σ has a strictly convex closure by Beardon [7], and for polytopes by Lins [34]. Further supporting evidence was obtained in [2,28,35,41].…”

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

“…If T has a fixed point x * in Σ , X is finite-dimensional, and K is polyhedral, ω(x; T ) is well understood; it is known (see [18]) that for each x ∈ Σ, ω(x; T ) is a periodic orbit of T and often (see [16]) equals {x * }. Compare, also, results in [1]. Note that if X is finite-dimensional, we can take q ∈ X * ; Σ is then convex and corresponds to D in Theorems 1.1 and 1.2.…”

A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains

“…This implies that ω(x; f ) is contained in a single part of C (see [16]), which is a stronger result than merely asserting that ω(x; f ) is contained in a convex subset of ∂ . This shows that order-preserving homogeneous of degree one maps can have more complicated dynamics than one might have expected from the results discussed in [1] and [11]. In Section 3 of this paper, we give an example of an order-preserving, homogeneous of degree one map on the standard cone in R n which, when scaled, can have omega limit sets that contain any convex subset of ∂ .…”