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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Chaotic functions with zero topological entropy
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by J. Smítal PDF
Trans. Amer. Math. Soc. 297 (1986), 269-282 Request permission

Abstract:

Recently Li and Yorke introduced the notion of chaos for mappings from the class ${C^0}(I,I)$, where $I$ is a compact real interval. In the present paper we give a characterization of the class $M \subset {C^0}(I,I)$ of mappings chaotic in this sense. As is well known, $M$ contains the mappings of positive topological entropy. We show that $M$ contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of $M$ consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) $\delta$-chaos, distinguishable on a certain level $\delta > 0$.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 297 (1986), 269-282
  • MSC: Primary 58F13; Secondary 54H20, 58F11
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0849479-9
  • MathSciNet review: 849479