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

 

Some properties of coupled-expanding maps in compact sets


Authors: Xu Zhang, Yuming Shi and Guanrong Chen
Journal: Proc. Amer. Math. Soc. 141 (2013), 585-595
MSC (2010): Primary 37D45; Secondary 37B10
Published electronically: June 19, 2012
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Abstract: In this paper, some properties of a strictly $ A$-coupled-expanding map in compact subsets of a metric space are studied, where $ A$ is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for $ A$. If the subshift for $ A$ has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for $ A$ and chaotic in the sense of Devaney.


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Additional Information

Xu Zhang
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
Email: xuzhang08@gmail.com

Yuming Shi
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
Email: ymshi@sdu.edu.cn

Guanrong Chen
Affiliation: Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, People’s Republic of China
Email: eegchen@cityu.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11339-5
PII: S 0002-9939(2012)11339-5
Keywords: Chaos, coupled-expanding map, transition matrix, topological entropy, topological semi-conjugacy
Received by editor(s): November 8, 2010
Received by editor(s) in revised form: July 2, 2011
Published electronically: June 19, 2012
Additional Notes: This research was partially supported by the RFDP of Higher Education of China (Grant 2010013 1110024) and the NNSF of China (Grant 11071143), the Hong Kong Research Council under Grant CityU 117/10E, and the Graduate Innovation Fund of Shandong University (Grant 11140070613137).
Communicated by: Yingfei Yi
Article copyright: © Copyright 2012 American Mathematical Society