Kybernetika 39 no. 2, 205-215, 2003

Continuous extension of order-preserving homogeneous maps

Andrew D. Burbanks, Colin T. Sparrow and Roger D. Nussbaum


Maps $f$ defined on the interior of the standard non-negative cone $K$ in ${\mathbb{R}}^N$ 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 $\partial{K}-\{0\}$.