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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. 157176. ISSN 03050041

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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 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Iterative methods (Mathematics), Homogeneous spaces, Cone, PolyhedraMathematical models, Geometry, Solid, Geometry, Descriptive 
Journal or Publication Title:  Mathematical Proceedings of the Cambridge Philosophical Society 
Publisher:  Cambridge University Press 
ISSN:  03050041 
Official Date:  January 2006 
Volume:  Vol.140 
Number:  No.1 
Page Range:  pp. 157176 
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) 
References:  [1] M. Akian and S. Gaubert. Spectral theorem for convex monotone homogeneous maps, and ergodic control. Nonlinear Anal. 52(2) (2003), 637–679. 
URI:  http://wrap.warwick.ac.uk/id/eprint/711 
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