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Iteration of order preserving subhomogeneous maps on a cone
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Akian, Marianne, Gaubert, S., Lemmens, Bas and Nussbaum, Roger D. (2006) Iteration of order preserving subhomogeneous maps on a cone. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.140 (No.1). pp. 157-176. doi:10.1017/S0305004105008832 ISSN 0305-0041.
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Official URL: http://dx.doi.org/10.1017/S0305004105008832
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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | January 2006 | ||||
Dates: |
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Volume: | Vol.140 | ||||
Number: | No.1 | ||||
Page Range: | pp. 157-176 | ||||
DOI: | 10.1017/S0305004105008832 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Funder: | Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Netherlands Organisation for Scientific Research] (NWO) |
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