Characterizations of weakly chaotic maps of the interval

Fedorenko, V. V.; Šarkovskii, A. N.; Smítal, J.

doi:10.1090/S0002-9939-1990-1017846-5

Characterizations of weakly chaotic maps of the interval
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by V. V. Fedorenko, A. N. Šarkovskii and J. Smítal PDF

Proc. Amer. Math. Soc. 110 (1990), 141-148 Request permission

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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