On the stability of chaotic functions. (English) Zbl 0639.26005

Janková, Katarína

Čas. Pěstování Mat. 112, 351-354 (1987).

Extremely chaotic functions and chaotic functions with very small scrambled sets are studied.
Let C be the class of all continuous functions f: [0,1]\(\to [0,1]\) and \(F\subset C\) the class of all chaotic functions. The following three subclasses of C are investigated:
\(F_ 1\)- the set of all chaotic functions possessing a scrambled set of positive outer Lebesgue measure,
\(F_ 2\)- the set of all chaotic functions possessing a scrambled set of positive Lebesgue measure,
\(F_ 3\)- the set of all chaotic functions possessing a scrambled set of the 2nd Baire category in [0,1].
It is shown that the sets \(F_ 1\), \(F_ 2\), \(F_ 3\) are dense in C and also the set \(F\setminus (F_ 2\cup F_ 3)\) is dense in C.

Reviewer: K.Janková

MSC:

26A18	Iteration of real functions in one variable
54H20	Topological dynamics (MSC2010)

Keywords:

stability; small perturbations; periodic point; chaotic functions; scrambled sets