Geodesic
Continuous extension of order-preserving homogeneous maps
Kybernetika, Tome 39 (2003) no. 2, p. [205]

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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-\lbrace 0\rbrace $. In the case where the cycle time $\chi (f)$ of the original map does not exist, such eigenvectors must lie in $\partial {K}-\lbrace 0\rbrace $.
Classification : 06F05, 47H07, 47N70, 93B27, 93B28, 93C65
Keywords: discrete event systems; order-preserving homogeneous maps 
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     title = {Continuous extension of order-preserving homogeneous maps},
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     pages = {[205]},
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     volume = {39},
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     zbl = {1249.93123},
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     url = {http://geodesic.mathdoc.fr/item/KYB_2003__39_2_a9/}
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Burbanks, Andrew D.; Sparrow, Colin T.; Nussbaum, Roger D. Continuous extension of order-preserving homogeneous maps. Kybernetika, Tome 39 (2003) no. 2, p. [205]. http://geodesic.mathdoc.fr/item/KYB_2003__39_2_a9/