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On the dynamics of sup-norm non-expansive maps
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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. doi:10.1017/S0143385704000665 ISSN 0143-3857.
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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 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | June 2005 | ||||
Dates: |
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Volume: | Vol.25 | ||||
Number: | No.3 | ||||
Page Range: | pp. 861-871 | ||||
DOI: | 10.1017/S0143385704000665 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Funder: | Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Netherlands Organisation for Scientific Research] (NWO) |
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