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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on periodic points of order preserving subhomogeneous maps


Authors: Bas Lemmens and Colin Sparrow
Journal: Proc. Amer. Math. Soc. 134 (2006), 1513-1517
MSC (2000): Primary 54H20, 47H07; Secondary 15A48, 46T20
Published electronically: October 7, 2005
MathSciNet review: 2199200
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Abstract: Let $ \mathbb{R}_+^n$ be the standard closed positive cone in $ \mathbb{R}^n$ and let $ \Gamma(\mathbb{R}_+^n)$ be the set of integers $ p\geq 1$ for which there exists a continuous, order preserving, subhomogeneous map $ f\colon \mathbb{R}_+^n\to \mathbb{R}_+^n$, which has a periodic point with period $ p$. It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that $ \Gamma(\mathbb{R}_+^n)$ is contained in the set $ B(n)$ consisting of those $ p\geq 1$ for which there exist integers $ q_1$ and $ q_2$ such that $ p=q_1q_2$, $ 1\leq q_1\leq {n\choose m}$, and $ 1\leq q_2\leq {m\choose \lfloor m/2\rfloor}$ for some $ 1\leq m\leq n$. This note shows that $ \Gamma(\mathbb{R}_+^n)=B(n)$ for all $ n\geq 1$.


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Additional Information

Bas Lemmens
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: lemmens@maths.warwick.ac.uk

Colin Sparrow
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: csparrow@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08390-5
PII: S 0002-9939(05)08390-5
Keywords: Monotone dynamical systems, nonlinear Perron-Frobenius theory
Received by editor(s): November 23, 2004
Published electronically: October 7, 2005
Additional Notes: The first author was supported by a TALENT-Fellowship of the Netherlands Organization for Scientific Research (NWO)
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.