Lemmens, Bas and Scheutzow, Michael. (2005) On the dynamics of sup-norm non-expansive maps. Ergodic Theory and Dynamical Systems, Vol.25 (No.3). pp. 861-871. ISSN 0143-3857

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Abstract

We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map $f\colon M\to M$, where $M\subset \mathbb{R}^n$, is at most $\max_k\, 2^k \big(\begin{smallmatrix}n\\ k\end{smallmatrix}\big)$. This upper bound is smaller than 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions $f\colon\mathbb{R}^n\to\mathbb{R}^n$ Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is $\big(\begin{smallmatrix}n\\ \lfloor n/2\rfloor\end{smallmatrix}\big)$. We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Combinatorial set theory, Mappings (Mathematics), Algebraic functions
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	June 2005
Volume:	Vol.25
Number:	No.3
Page Range:	pp. 861-871
Identification Number:	10.1017/S0143385704000665
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Netherlands Organisation for Scientific Research] (NWO)
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URI:	http://wrap.warwick.ac.uk/id/eprint/744

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