TY - JOUR

T1 - Extension of order-preserving maps on a cone

AU - Burbanks, Andrew D.

AU - Nussbaum, Roger D.

AU - Sparrow, Colin T.

N1 - Funding Information:
A.D.B. was supported partly by EPSRC GR/L42742 and BRIMS (Hewletacktard-P Laboratories, Bristol), and by the European research newotrk ALES A(etPh ED `Algebraic Approach to PermranfceoEaluavtion of Discrete Eet vnSyms’).ste R.D.N. was supported partly by NSF DMS 0070829.

PY - 2003

Y1 - 2003

N2 - 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 δK - {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.

AB - 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 δK - {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.

UR - http://www.scopus.com/inward/record.url?scp=0037229434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037229434&partnerID=8YFLogxK

U2 - 10.1017/s0308210500002274

DO - 10.1017/s0308210500002274

M3 - Article

AN - SCOPUS:0037229434

VL - 133

SP - 35

EP - 59

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1

ER -