Iteration of order preserving subhomogeneous maps on a cone

Lemmens, Bas and Akian, Marianne and Gaubert, Stephane and Nussbaum, Roger (2006) Iteration of order preserving subhomogeneous maps on a cone. Mathematical Proceedings of the Cambridge Philosophical Society, 140 (1). pp. 157-176. ISSN 0305-0041. (The full text of this publication is not available from this repository)

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http://dx.doi.org.chain.kent.ac.uk/10.1017/S030500...

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: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Divisions: Central Services
Depositing User: Bas Lemmens
Date Deposited: 17 Nov 2011 15:30
Last Modified: 30 Apr 2014 10:01
Resource URI: http://kar.kent.ac.uk/id/eprint/28438 (The current URI for this page, for reference purposes)
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