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Errata: Typical continuous functions are not chaotic in the sense of Devaney. (English) Zbl 1012.37010

The author writes: “In this note, necessary changes, clarifications and corrections are given to the results of [the author, ibid. 25, No. 2, 947-953 (2000; Zbl 1012.37009), reviewed above]”.
Instead of \(C(M,M)\), where \(M\) is a compact metric space, in this note the set \(C(K,K)\) of continuous functions \(f:K\to K\), where \(K\) is the Cartesian product of \(n\) compact intervals in the real line with usual metric \(d\), is considered. The set \(C(K,K)\) has the metric \(D(f,h)= \sup\{d(f(x), h(x)): x\in K\}\). Results of the note:
1. There exists a dense open subset \(W\) in \(C(K,K)\) such that every function in \(W\) is not topologically transitive, and hence not chaotic in the sense of Devaney. 2. There exists a dense open subset \(W\) in \(C(K,K)\) such that every function in \(W\) has an asymptotically stable set. 3. There exists a dense open subset \(W\) in \(C(K,K)\) such that every function \(f\) in \(W\) has the property that \(CR(f)\neq K\), where \(CR(f)\) denotes the chain recurrent set of \(f\). 4. Let \(P(I,I)\) denote the set of all polynomials on the compact interval \(I\). There exists dense open subset \(S\subset P(I,I)\) such that each polynomial in \(S\) is Block-Coppel chaotic but not chaotic in the sense of Devaney.

MSC:

37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable

Citations:

Zbl 1012.37009
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