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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterizations of weakly chaotic maps of the interval


Authors: V. V. Fedorenko, A. N. Šarkovskii and J. Smítal
Journal: Proc. Amer. Math. Soc. 110 (1990), 141-148
MSC: Primary 58F20; Secondary 28D20, 54H20, 58F08, 58F13
DOI: https://doi.org/10.1090/S0002-9939-1990-1017846-5
MathSciNet review: 1017846
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Abstract: We prove, among others, the following relations between notions of chaos for continuous maps of the interval: (i) A map $ f$ is not chaotic in the sense of Li and Yorke iff $ f$ restricted to the set of its $ \omega $-limit points is stable in the sense of Ljapunov. (ii) The topological entropy of $ f$ is zero iff $ f$ restricted to the set of chain recurrent points is not chaotic in the sense of Li and Yorke, and this is iff every trajectory is approximable by trajectories of periodic intervals.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1017846-5
Article copyright: © Copyright 1990 American Mathematical Society