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Transitive sensitive subsystems for interval maps. (English) Zbl 1094.37020

This paper studies the discrete dynamical system induced by iteration of a continuous map \(f:I \rightarrow I\), where \(I\) is a compact interval. In this setting, connections are made between several concepts related to chaos. For a continuous map \(T:X \rightarrow X\), where \(X\) is a metric space, \(T\) is said to be chaotic in the sense of Wiggins if there exists a nonempty closed invariant subset \(Y\) such that \(T| _Y\) is transitive and has sensitive dependence on initial conditions. The aim of this paper is to relate chaos in the sense of Wiggins to some of the other concepts related to chaos. Among the main results are:
(1) There exists a continuous map \(f:I \rightarrow I\) with zero topological entropy which is chaotic in the sense of Wiggins.
(2) For \(f:I \rightarrow I\), if \(f\) is chaotic in the sense of Wiggins, then it is chaotic in the sense of Li-Yorke.
(3) There exists a continuous map \(g:I \rightarrow I\) which is chaotic in the sense of Li-Yorke but not chaotic in the sense of Wiggins.
To summarize, “for interval maps, chaos in the sense of Wiggins is a strictly intermediate notion between positive topological entropy and chaos in the sense of Li-Yorke”. See also S. F. Kolyada [Ukr. Mat. Zh. 56, No. 8, 1043–1061 (2004; Zbl 1075.37500)].

MSC:

37E05 Dynamical systems involving maps of the interval
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Citations:

Zbl 1075.37500
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