# 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. (doi:https://doi.org/10.1017/S0305004105008832) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

 The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication) Official URLhttp://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 Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Central Services Bas Lemmens 17 Nov 2011 15:30 UTC 30 Apr 2014 10:01 UTC https://kar.kent.ac.uk/id/eprint/28438 (The current URI for this page, for reference purposes)