Iteration of order preserving subhomogeneous maps on a cone

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps that map a polyhedral cone into itself. For these maps we show that every bounded orbit converges to a periodic orbit and, moreover, that there exists an a priori upper bound for the periods of periodic points that only depends on the number of facets of the polyhedral cone. By constructing examples on the standard positive cone, we show that the upper bound is asymptotically sharp. These results are an extension of recent work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone.