Akian, Marianne, Gaubert, S. (Stephane), Lemmens, Bas and Nussbaum, Roger D., 1944-. (2006) Iteration of order preserving subhomogeneous maps on a cone. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.14 (No.1). pp. 157-176. ISSN 0305-0041

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Abstract

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K\,{\rightarrow}\, K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by \[ \beta_N = \max_{q+r+s=N}\frac{N!}{q!r!s!}= \frac{N!}{\big\lfloor\frac{N}{3}\big\rfloor!\big\lfloor\frac{N\,{+}\,1}{3}\big\rfloor! \big\lfloor\frac{N\,{+}\,2}{3}\big\rfloor!}\sim \frac{3^{N+1}\sqrt{3}}{2\pi N}, \] where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in $\mathbb{R}^n$, we show that the upper bound is asymptotically sharp.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Iterative methods (Mathematics), Homogeneous spaces, Cone, Polyhedra--Mathematical models, Geometry, Solid, Geometry, Descriptive
Journal or Publication Title:	Mathematical Proceedings of the Cambridge Philosophical Society
Publisher:	Cambridge University Press
ISSN:	0305-0041
Date:	January 2006
Volume:	Vol.14
Number:	No.1
Page Range:	pp. 157-176
Identification Number:	10.1017/S0305004105008832
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Netherlands Organisation for Scientific Research] (NWO)
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URI:	http://wrap.warwick.ac.uk/id/eprint/711

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