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:10.1017/S0305004105008832)
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Official URL 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 |
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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: | https://kar.kent.ac.uk/id/eprint/28438 (The current URI for this page, for reference purposes) |

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