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There are no chaotic mappings with residual scrambled sets. (English) Zbl 0646.26008
Let f be a continuous map of a compact interval I to itself. Let \(S\subset I\) and let for any x,y\(\in S\), \(x\neq y\), \(\limsup | f\quad n(x)-f\quad n(y)| >0\) and \(\liminf | f\quad n(x)-f\quad n(y)| =0\) for \(n\to \infty\), where f n denotes the n-th iterate of f. Then S is residual in no subinterval of I.
Reviewer: J.Smítal

26A18 Iteration of real functions in one variable
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54H20 Topological dynamics (MSC2010)
Full Text: DOI
[1] DOI: 10.2307/2000468 · Zbl 0639.54029
[2] DOI: 10.2307/2045151 · Zbl 0592.26006
[3] DOI: 10.2307/2044350 · Zbl 0555.26003
[4] Janková, Bull. Auetral. Math. Soc. 34 pp 283– (1986)
[5] DOI: 10.2307/2318254 · Zbl 0351.92021
[6] DOI: 10.2307/2045150 · Zbl 0592.26005
[7] DOI: 10.1007/BFb0076425
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