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Characterizations of weakly chaotic maps of the interval. (English) Zbl 0728.26008
Let C(I,I) be the space of continuous maps f: \(I\to I\), I a real compact interval. For any \(f\in C(I,I)\) \(f^ n\) denotes the n-th iterate of f, \(\omega_ f(x)\) is the \(\omega\)-limit set of x, \(\omega =\cup \{\omega_ f(x)\},\quad x\in I,\) is the \(\omega\)-limit set of f. A point \(x\in I\) is chain recurrent if for any \(\epsilon >0\) there is a sequence \(\{x_ i\}^ n_{i=0}\) with \(x_ 0=x_ n=x\) and \(| f(x_ i)- f(x_{i+1})| <\epsilon\) for any \(i<n\). The considered chaos is the chaos in the sense of Li and Yorke and \(f\in C(I,I)\) is referred to as weakly chaotic if it is chaotic with zero topological entropy. In the paper, relations between the above notions and notions as stability in the sense of Lyapunov or approximability of trajectories by periodic points are studied. It is shown, among others, that 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 Lyapunov. Further 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.

MSC:
26A18 Iteration of real functions in one variable
37B99 Topological dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54H20 Topological dynamics (MSC2010)
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