Continuous extension of order-preserving homogeneous maps

Maps f defined on the interior of the standard non-negative cone K in R^N which are both homogeneous of degree 1 and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson's part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that the extension must have at least one eigenvector in K - {0}. In the case where the cycle time chi(f) of the original map does not exist, such eigenvectors must lie in the boundary of K - {0}.