Skip to main content
    • Aa
    • Aa

Extension of order-preserving maps on a cone

  • Andrew D. Burbanks (a1), Roger D. Nussbaum (a2) and Colin T. Sparrow (a3)

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.

We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous.

Maps f defined on the interior of the non-negative cone K in RN, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K – {0}. In the case where the cycle time χ(f) of the original map does not exist, such eigenvectors must lie in ∂K – {0}.

We conclude with some discussions and applications to operator-valued means. We also extend our results to an ‘intermediate’ situation, which arises in some important application areas, particularly in the construction of diffusions on certain fractals via maps defined on the interior of cones of Dirichlet forms.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 56 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st October 2017. This data will be updated every 24 hours.