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On the dynamics of supnorm nonexpansive maps
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Lemmens, Bas and Scheutzow, Michael. (2005) On the dynamics of supnorm nonexpansive maps. Ergodic Theory and Dynamical Systems, Vol.25 (No.3). pp. 861871. ISSN 01433857

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Official URL: http://dx.doi.org/10.1017/S0143385704000665
Abstract
We present several results for the periods of periodic points of supnorm nonexpansive maps. In particular, we show that the period of each periodic point of a supnorm nonexpansive 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 supnorm nonexpansive 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 antichains 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:  01433857 
Official Date:  June 2005 
Volume:  Vol.25 
Number:  No.3 
Page Range:  pp. 861871 
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) 
References:  [1] M. A. Akcoglu and U. Krengel. Nonlinear models of diffusion on a finite space. Probab. Theory Related Fields 76(4) (1987), 411–420. 
URI:  http://wrap.warwick.ac.uk/id/eprint/744 
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